Calculus Examples

$\frac{| 4 - x^{2} |}{x - 2}$

Step 1

Write $\frac{| 4 - x^{2} |}{x - 2}$ as a function.

$f (x) = \frac{| 4 - x^{2} |}{x - 2}$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = | 4 - x^{2} |$ and $g (x) = x - 2$ .

$f' (x) = \frac{(x - 2) \frac{d}{d x} (| 4 - x^{2} |) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = 4 - x^{2}$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u$ as $4 - x^{2}$ .

$f' (x) = \frac{(x - 2) (\frac{d}{d u} (| u |) \frac{d}{d x} (4 - x^{2})) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.2.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$f' (x) = \frac{(x - 2) (\frac{u}{| u |} \cdot \frac{d}{d x} (4 - x^{2})) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.2.3

Replace all occurrences of $u$ with $4 - x^{2}$ .

$f' (x) = \frac{(x - 2) (\frac{4 - x^{2}}{| 4 - x^{2} |} \cdot \frac{d}{d x} (4 - x^{2})) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

By the Sum Rule, the derivative of $4 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]$ .

$f' (x) = \frac{(x - 2) (\frac{4 - x^{2}}{| 4 - x^{2} |}) \cdot (\frac{d}{d x} (4) + \frac{d}{d x} (- x^{2})) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f' (x) = \frac{(x - 2) (\frac{4 - x^{2}}{| 4 - x^{2} |}) \cdot (0 + \frac{d}{d x} (- x^{2})) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3.3

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$f' (x) = \frac{(x - 2) (\frac{4 - x^{2}}{| 4 - x^{2} |}) \cdot \frac{d}{d x} (- x^{2}) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$f' (x) = \frac{(x - 2) (\frac{4 - x^{2}}{| 4 - x^{2} |}) \cdot (- \frac{d}{d x} x^{2}) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = \frac{(x - 2) (\frac{4 - x^{2}}{| 4 - x^{2} |}) \cdot (- (2 x)) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3.6

Combine fractions.

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Step 2.1.3.6.1

Multiply $2$ by $- 1$ .

$f' (x) = \frac{(x - 2) (\frac{4 - x^{2}}{| 4 - x^{2} |}) \cdot (- 2 x) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3.6.2

Combine $- 2$ and $\frac{4 - x^{2}}{| 4 - x^{2} |}$ .

$f' (x) = \frac{(x - 2) (\frac{- 2 (4 - x^{2})}{| 4 - x^{2} |}) \cdot x - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3.6.3

Combine $x$ and $\frac{- 2 (4 - x^{2})}{| 4 - x^{2} |}$ .

$f' (x) = \frac{(x - 2) (\frac{x (- 2 (4 - x^{2}))}{| 4 - x^{2} |}) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3.6.4

Simplify the expression.

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Step 2.1.3.6.4.1

Move the negative in front of the fraction.

$f' (x) = \frac{(x - 2) (- \frac{x \cdot (2 (4 - x^{2}))}{| 4 - x^{2} |}) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3.6.4.2

Move $2$ to the left of $x$ .

$f' (x) = \frac{(x - 2) (- \frac{2 \cdot (x (4 - x^{2}))}{| 4 - x^{2} |}) - | 4 - x^{2} | \frac{d}{d x} (x - 2)}{{(x - 2)}^{2}}$

Step 2.1.3.7

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (4 - x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} | (\frac{d}{d x} (x) + \frac{d}{d x} (- 2))}{{(x - 2)}^{2}}$

Step 2.1.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (4 - x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} | (1 + \frac{d}{d x} (- 2))}{{(x - 2)}^{2}}$

Step 2.1.3.9

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (4 - x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} | (1 + 0)}{{(x - 2)}^{2}}$

Step 2.1.3.10

Simplify the expression.

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Step 2.1.3.10.1

Add $1$ and $0$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (4 - x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} | \cdot 1}{{(x - 2)}^{2}}$

Step 2.1.3.10.2

Multiply $- 1$ by $1$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (4 - x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Apply the distributive property.

$f' (x) = \frac{(x - 2) (- \frac{2 x \cdot 4 + 2 x (- x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2

Simplify the numerator.

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Step 2.1.4.2.1

Simplify each term.

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Step 2.1.4.2.1.1

Simplify the numerator.

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Step 2.1.4.2.1.1.1

Multiply $4$ by $2$ .

$f' (x) = \frac{(x - 2) (- \frac{8 x + 2 x (- x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.2

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{(x - 2) (- \frac{8 x + 2 \cdot (- 1 x \cdot x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.1.4.2.1.1.3.1

Move $x^{2}$ .

$f' (x) = \frac{(x - 2) (- \frac{8 x + 2 \cdot (- 1 (x^{2} x))}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.3.2

Multiply $x^{2}$ by $x$ .

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Step 2.1.4.2.1.1.3.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{(x - 2) (- \frac{8 x + 2 \cdot (- 1 (x^{2} x))}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{(x - 2) (- \frac{8 x + 2 \cdot (- 1 x^{2 + 1})}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.3.3

Add $2$ and $1$ .

$f' (x) = \frac{(x - 2) (- \frac{8 x + 2 \cdot (- 1 x^{3})}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.4

Multiply $2$ by $- 1$ .

$f' (x) = \frac{(x - 2) (- \frac{8 x - 2 x^{3}}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.5

Rewrite $8 x - 2 x^{3}$ in a factored form.

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Step 2.1.4.2.1.1.5.1

Factor $2 x$ out of $8 x - 2 x^{3}$ .

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Step 2.1.4.2.1.1.5.1.1

Factor $2 x$ out of $8 x$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (4) - 2 x^{3}}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.5.1.2

Factor $2 x$ out of $- 2 x^{3}$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (4) + 2 x (- x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.5.1.3

Factor $2 x$ out of $2 x (4) + 2 x (- x^{2})$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (4 - x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.5.2

Rewrite $4$ as $2^{2}$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2^{2} - x^{2})}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.1.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2

Simplify the denominator.

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Step 2.1.4.2.1.2.1

Rewrite $4$ as $2^{2}$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 2^{2} - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.3

Expand $(2 + x) (2 - x)$ using the FOIL Method.

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Step 2.1.4.2.1.2.3.1

Apply the distributive property.

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 2 (2 - x) + x (2 - x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.3.2

Apply the distributive property.

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 2 \cdot 2 + 2 (- x) + x (2 - x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.3.3

Apply the distributive property.

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 2 \cdot 2 + 2 (- x) + x \cdot 2 + x (- x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.4

Simplify and combine like terms.

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Step 2.1.4.2.1.2.4.1

Simplify each term.

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Step 2.1.4.2.1.2.4.1.1

Multiply $2$ by $2$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 4 + 2 (- x) + x \cdot 2 + x (- x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.4.1.2

Multiply $- 1$ by $2$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 4 - 2 x + x \cdot 2 + x (- x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.4.1.3

Move $2$ to the left of $x$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 4 - 2 x + 2 \cdot x + x (- x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.4.1.4

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 4 - 2 x + 2 x - x \cdot x |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.4.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.4.2.1.2.4.1.5.1

Move $x$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 4 - 2 x + 2 x - (x \cdot x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.4.1.5.2

Multiply $x$ by $x$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 4 - 2 x + 2 x - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.4.2

Add $- 2 x$ and $2 x$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 4 + 0 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.4.3

Add $4$ and $0$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 4 - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.5

Rewrite $4$ as $2^{2}$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| 2^{2} - x^{2} |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.2.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x$ .

$f' (x) = \frac{(x - 2) (- \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.3

Apply the distributive property.

$f' (x) = \frac{x (- \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |}) - 2 (- \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.4

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{- x \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - 2 (- \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.5

Multiply $- 2 (- \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |})$ .

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Step 2.1.4.2.1.5.1

Multiply $- 1$ by $- 2$ .

$f' (x) = \frac{- x \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} + 2 (\frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |}) - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.5.2

Combine $2$ and $\frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |}$ .

$f' (x) = \frac{- x \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} + \frac{2 (2 x (2 + x) (2 - x))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.5.3

Multiply $2$ by $2$ .

$f' (x) = \frac{- x \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} + \frac{4 ((x (2 + x)) (2 - x))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.6

Multiply $- x \frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |}$ .

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Step 2.1.4.2.1.6.1

Combine $\frac{2 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |}$ and $x$ .

$f' (x) = \frac{- \frac{2 x (2 + x) (2 - x) x}{| (2 + x) (2 - x) |} + \frac{4 ((x (2 + x)) (2 - x))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.6.2

Raise $x$ to the power of $1$ .

$f' (x) = \frac{- \frac{2 (x \cdot x) (2 + x) (2 - x)}{| (2 + x) (2 - x) |} + \frac{4 ((x (2 + x)) (2 - x))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.6.3

Raise $x$ to the power of $1$ .

$f' (x) = \frac{- \frac{2 (x \cdot x) (2 + x) (2 - x)}{| (2 + x) (2 - x) |} + \frac{4 ((x (2 + x)) (2 - x))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{- \frac{2 x^{1 + 1} (2 + x) (2 - x)}{| (2 + x) (2 - x) |} + \frac{4 ((x (2 + x)) (2 - x))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.6.5

Add $1$ and $1$ .

$f' (x) = \frac{- \frac{2 x^{2} (2 + x) (2 - x)}{| (2 + x) (2 - x) |} + \frac{4 ((x (2 + x)) (2 - x))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

$f' (x) = \frac{- \frac{2 x^{2} (2 + x) (2 - x)}{| (2 + x) (2 - x) |} + \frac{4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.7

Combine the numerators over the common denominator.

$f' (x) = \frac{\frac{- 2 x^{2} (2 + x) (2 - x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8

Simplify each term.

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Step 2.1.4.2.1.8.1

Apply the distributive property.

$f' (x) = \frac{\frac{(- 2 x^{2} \cdot 2 - 2 x^{2} x) (2 - x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.2

Multiply $2$ by $- 2$ .

$f' (x) = \frac{\frac{(- 4 x^{2} - 2 x^{2} x) (2 - x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.4.2.1.8.3.1

Move $x$ .

$f' (x) = \frac{\frac{(- 4 x^{2} - 2 (x \cdot x^{2})) (2 - x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.3.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.4.2.1.8.3.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{\frac{(- 4 x^{2} - 2 (x \cdot x^{2})) (2 - x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{\frac{(- 4 x^{2} - 2 x^{1 + 2}) (2 - x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.3.3

Add $1$ and $2$ .

$f' (x) = \frac{\frac{(- 4 x^{2} - 2 x^{3}) (2 - x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.4

Expand $(- 4 x^{2} - 2 x^{3}) (2 - x)$ using the FOIL Method.

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Step 2.1.4.2.1.8.4.1

Apply the distributive property.

$f' (x) = \frac{\frac{- 4 x^{2} (2 - x) - 2 x^{3} (2 - x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.4.2

Apply the distributive property.

$f' (x) = \frac{\frac{- 4 x^{2} \cdot 2 - 4 x^{2} (- x) - 2 x^{3} (2 - x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.4.3

Apply the distributive property.

$f' (x) = \frac{\frac{- 4 x^{2} \cdot 2 - 4 x^{2} (- x) - 2 x^{3} \cdot 2 - 2 x^{3} (- x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5

Simplify and combine like terms.

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Step 2.1.4.2.1.8.5.1

Simplify each term.

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Step 2.1.4.2.1.8.5.1.1

Multiply $2$ by $- 4$ .

$f' (x) = \frac{\frac{- 8 x^{2} - 4 x^{2} (- x) - 2 x^{3} \cdot 2 - 2 x^{3} (- x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.2

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{\frac{- 8 x^{2} - 4 \cdot (- 1 x^{2} x) - 2 x^{3} \cdot 2 - 2 x^{3} (- x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.4.2.1.8.5.1.3.1

Move $x$ .

$f' (x) = \frac{\frac{- 8 x^{2} - 4 \cdot (- 1 (x \cdot x^{2})) - 2 x^{3} \cdot 2 - 2 x^{3} (- x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.4.2.1.8.5.1.3.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{\frac{- 8 x^{2} - 4 \cdot (- 1 (x \cdot x^{2})) - 2 x^{3} \cdot 2 - 2 x^{3} (- x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{\frac{- 8 x^{2} - 4 \cdot (- 1 x^{1 + 2}) - 2 x^{3} \cdot 2 - 2 x^{3} (- x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.3.3

Add $1$ and $2$ .

$f' (x) = \frac{\frac{- 8 x^{2} - 4 \cdot (- 1 x^{3}) - 2 x^{3} \cdot 2 - 2 x^{3} (- x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.4

Multiply $- 4$ by $- 1$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 4 x^{3} - 2 x^{3} \cdot 2 - 2 x^{3} (- x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.5

Multiply $2$ by $- 2$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 4 x^{3} - 4 x^{3} - 2 x^{3} (- x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.6

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{\frac{- 8 x^{2} + 4 x^{3} - 4 x^{3} - 2 \cdot (- 1 x^{3} x) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.7

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.1.4.2.1.8.5.1.7.1

Move $x$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 4 x^{3} - 4 x^{3} - 2 \cdot (- 1 (x \cdot x^{3})) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.7.2

Multiply $x$ by $x^{3}$ .

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Step 2.1.4.2.1.8.5.1.7.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 4 x^{3} - 4 x^{3} - 2 \cdot (- 1 (x \cdot x^{3})) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{\frac{- 8 x^{2} + 4 x^{3} - 4 x^{3} - 2 \cdot (- 1 x^{1 + 3}) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.7.3

Add $1$ and $3$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 4 x^{3} - 4 x^{3} - 2 \cdot (- 1 x^{4}) + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.1.8

Multiply $- 2$ by $- 1$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 4 x^{3} - 4 x^{3} + 2 x^{4} + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.2

Subtract $4 x^{3}$ from $4 x^{3}$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 0 + 2 x^{4} + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.5.3

Add $- 8 x^{2}$ and $0$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 4 x (2 + x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.6

Apply the distributive property.

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + (4 x \cdot 2 + 4 x \cdot x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.7

Multiply $2$ by $4$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + (8 x + 4 x \cdot x) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.8

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.4.2.1.8.8.1

Move $x$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + (8 x + 4 (x \cdot x)) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.8.2

Multiply $x$ by $x$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + (8 x + 4 x^{2}) (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.9

Expand $(8 x + 4 x^{2}) (2 - x)$ using the FOIL Method.

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Step 2.1.4.2.1.8.9.1

Apply the distributive property.

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 8 x (2 - x) + 4 x^{2} (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.9.2

Apply the distributive property.

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 8 x \cdot 2 + 8 x (- x) + 4 x^{2} (2 - x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.9.3

Apply the distributive property.

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 8 x \cdot 2 + 8 x (- x) + 4 x^{2} \cdot 2 + 4 x^{2} (- x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10

Simplify and combine like terms.

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Step 2.1.4.2.1.8.10.1

Simplify each term.

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Step 2.1.4.2.1.8.10.1.1

Multiply $2$ by $8$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x + 8 x (- x) + 4 x^{2} \cdot 2 + 4 x^{2} (- x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.2

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x + 8 \cdot (- 1 x \cdot x) + 4 x^{2} \cdot 2 + 4 x^{2} (- x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.4.2.1.8.10.1.3.1

Move $x$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x + 8 \cdot (- 1 (x \cdot x)) + 4 x^{2} \cdot 2 + 4 x^{2} (- x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.3.2

Multiply $x$ by $x$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x + 8 \cdot (- 1 x^{2}) + 4 x^{2} \cdot 2 + 4 x^{2} (- x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.4

Multiply $8$ by $- 1$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x - 8 x^{2} + 4 x^{2} \cdot 2 + 4 x^{2} (- x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.5

Multiply $2$ by $4$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x - 8 x^{2} + 8 x^{2} + 4 x^{2} (- x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.6

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x - 8 x^{2} + 8 x^{2} + 4 \cdot (- 1 x^{2} x)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.7

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.4.2.1.8.10.1.7.1

Move $x$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x - 8 x^{2} + 8 x^{2} + 4 \cdot (- 1 (x \cdot x^{2}))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.7.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.4.2.1.8.10.1.7.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x - 8 x^{2} + 8 x^{2} + 4 \cdot (- 1 (x \cdot x^{2}))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x - 8 x^{2} + 8 x^{2} + 4 \cdot (- 1 x^{1 + 2})}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.7.3

Add $1$ and $2$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x - 8 x^{2} + 8 x^{2} + 4 \cdot (- 1 x^{3})}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.1.8

Multiply $4$ by $- 1$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x - 8 x^{2} + 8 x^{2} - 4 x^{3}}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.2

Add $- 8 x^{2}$ and $8 x^{2}$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x + 0 - 4 x^{3}}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.8.10.3

Add $16 x$ and $0$ .

$f' (x) = \frac{\frac{- 8 x^{2} + 2 x^{4} + 16 x - 4 x^{3}}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9

Simplify the numerator.

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Step 2.1.4.2.1.9.1

Factor $2 x$ out of $- 8 x^{2} + 2 x^{4} + 16 x - 4 x^{3}$ .

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Step 2.1.4.2.1.9.1.1

Factor $2 x$ out of $- 8 x^{2}$ .

$f' (x) = \frac{\frac{2 x (- 4 x) + 2 x^{4} + 16 x - 4 x^{3}}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.1.2

Factor $2 x$ out of $2 x^{4}$ .

$f' (x) = \frac{\frac{2 x (- 4 x) + 2 x (x^{3}) + 16 x - 4 x^{3}}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.1.3

Factor $2 x$ out of $16 x$ .

$f' (x) = \frac{\frac{2 x (- 4 x) + 2 x (x^{3}) + 2 x (8) - 4 x^{3}}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.1.4

Factor $2 x$ out of $- 4 x^{3}$ .

$f' (x) = \frac{\frac{2 x (- 4 x) + 2 x (x^{3}) + 2 x (8) + 2 x (- 2 x^{2})}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.1.5

Factor $2 x$ out of $2 x (- 4 x) + 2 x (x^{3})$ .

$f' (x) = \frac{\frac{2 x (- 4 x + x^{3}) + 2 x (8) + 2 x (- 2 x^{2})}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.1.6

Factor $2 x$ out of $2 x (- 4 x + x^{3}) + 2 x (8)$ .

$f' (x) = \frac{\frac{2 x (- 4 x + x^{3} + 8) + 2 x (- 2 x^{2})}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.1.7

Factor $2 x$ out of $2 x (- 4 x + x^{3} + 8) + 2 x (- 2 x^{2})$ .

$f' (x) = \frac{\frac{2 x (- 4 x + x^{3} + 8 - 2 x^{2})}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.2

Reorder terms.

$f' (x) = \frac{\frac{2 x (x^{3} - 2 x^{2} - 4 x + 8)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.3

Factor out the greatest common factor from each group.

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Step 2.1.4.2.1.9.3.1

Group the first two terms and the last two terms.

$f' (x) = \frac{\frac{2 x ((x^{3} - 2 x^{2}) - 4 x + 8)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.3.2

Factor out the greatest common factor (GCF) from each group.

$f' (x) = \frac{\frac{2 x (x^{2} (x - 2) - 4 (x - 2))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.4

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$f' (x) = \frac{\frac{2 x ((x - 2) (x^{2} - 4))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.5

Rewrite $4$ as $2^{2}$ .

$f' (x) = \frac{\frac{2 x ((x - 2) (x^{2} - 2^{2}))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$f' (x) = \frac{\frac{2 x ((x - 2) (x + 2) (x - 2))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.7

Combine exponents.

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Step 2.1.4.2.1.9.7.1

Raise $x - 2$ to the power of $1$ .

$f' (x) = \frac{\frac{2 x ((x - 2) (x - 2) (x + 2))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.7.2

Raise $x - 2$ to the power of $1$ .

$f' (x) = \frac{\frac{2 x ((x - 2) (x - 2) (x + 2))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{\frac{2 x ({(x - 2)}^{1 + 1} (x + 2))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.1.9.7.4

Add $1$ and $1$ .

$f' (x) = \frac{\frac{2 x ({(x - 2)}^{2} (x + 2))}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

$f' (x) = \frac{\frac{2 x {(x - 2)}^{2} (x + 2)}{| (2 + x) (2 - x) |} - | 4 - x^{2} |}{{(x - 2)}^{2}}$

Step 2.1.4.2.2

To write $- | 4 - x^{2} |$ as a fraction with a common denominator, multiply by $\frac{| (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}$ .

$f' (x) = \frac{\frac{2 x {(x - 2)}^{2} (x + 2)}{| (2 + x) (2 - x) |} - | 4 - x^{2} | \cdot \frac{| (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.3

Combine $- | 4 - x^{2} |$ and $\frac{| (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}$ .

$f' (x) = \frac{\frac{2 x {(x - 2)}^{2} (x + 2)}{| (2 + x) (2 - x) |} + \frac{- | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.4

Combine the numerators over the common denominator.

$f' (x) = \frac{\frac{2 x {(x - 2)}^{2} (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5

Simplify the numerator.

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Step 2.1.4.2.5.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$f' (x) = \frac{\frac{2 x ((x - 2) (x - 2)) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 2.1.4.2.5.2.1

Apply the distributive property.

$f' (x) = \frac{\frac{2 x (x (x - 2) - 2 (x - 2)) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.2.2

Apply the distributive property.

$f' (x) = \frac{\frac{2 x (x \cdot x + x \cdot - 2 - 2 (x - 2)) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.2.3

Apply the distributive property.

$f' (x) = \frac{\frac{2 x (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.3

Simplify and combine like terms.

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Step 2.1.4.2.5.3.1

Simplify each term.

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Step 2.1.4.2.5.3.1.1

Multiply $x$ by $x$ .

$f' (x) = \frac{\frac{2 x (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.3.1.2

Move $- 2$ to the left of $x$ .

$f' (x) = \frac{\frac{2 x (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.3.1.3

Multiply $- 2$ by $- 2$ .

$f' (x) = \frac{\frac{2 x (x^{2} - 2 x - 2 x + 4) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.3.2

Subtract $2 x$ from $- 2 x$ .

$f' (x) = \frac{\frac{2 x (x^{2} - 4 x + 4) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.4

Apply the distributive property.

$f' (x) = \frac{\frac{(2 x \cdot x^{2} + 2 x (- 4 x) + 2 x \cdot 4) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.5

Simplify.

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Step 2.1.4.2.5.5.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.1.4.2.5.5.1.1

Move $x^{2}$ .

$f' (x) = \frac{\frac{(2 (x^{2} x) + 2 x (- 4 x) + 2 x \cdot 4) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.5.1.2

Multiply $x^{2}$ by $x$ .

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Step 2.1.4.2.5.5.1.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{\frac{(2 (x^{2} x) + 2 x (- 4 x) + 2 x \cdot 4) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.5.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{\frac{(2 x^{2 + 1} + 2 x (- 4 x) + 2 x \cdot 4) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.5.1.3

Add $2$ and $1$ .

$f' (x) = \frac{\frac{(2 x^{3} + 2 x (- 4 x) + 2 x \cdot 4) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.5.2

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{\frac{(2 x^{3} + 2 \cdot (- 4 x \cdot x) + 2 x \cdot 4) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.5.3

Multiply $4$ by $2$ .

$f' (x) = \frac{\frac{(2 x^{3} + 2 \cdot (- 4 x \cdot x) + 8 x) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.6

Simplify each term.

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Step 2.1.4.2.5.6.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.4.2.5.6.1.1

Move $x$ .

$f' (x) = \frac{\frac{(2 x^{3} + 2 \cdot (- 4 (x \cdot x)) + 8 x) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.6.1.2

Multiply $x$ by $x$ .

$f' (x) = \frac{\frac{(2 x^{3} + 2 \cdot (- 4 x^{2}) + 8 x) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.6.2

Multiply $2$ by $- 4$ .

$f' (x) = \frac{\frac{(2 x^{3} - 8 x^{2} + 8 x) (x + 2) - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.7

Expand $(2 x^{3} - 8 x^{2} + 8 x) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$f' (x) = \frac{\frac{2 x^{3} x + 2 x^{3} \cdot 2 - 8 x^{2} x - 8 x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8

Simplify each term.

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Step 2.1.4.2.5.8.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.1.4.2.5.8.1.1

Move $x$ .

$f' (x) = \frac{\frac{2 (x \cdot x^{3}) + 2 x^{3} \cdot 2 - 8 x^{2} x - 8 x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.1.2

Multiply $x$ by $x^{3}$ .

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Step 2.1.4.2.5.8.1.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{\frac{2 (x \cdot x^{3}) + 2 x^{3} \cdot 2 - 8 x^{2} x - 8 x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{\frac{2 x^{1 + 3} + 2 x^{3} \cdot 2 - 8 x^{2} x - 8 x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.1.3

Add $1$ and $3$ .

$f' (x) = \frac{\frac{2 x^{4} + 2 x^{3} \cdot 2 - 8 x^{2} x - 8 x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.2

Multiply $2$ by $2$ .

$f' (x) = \frac{\frac{2 x^{4} + 4 x^{3} - 8 x^{2} x - 8 x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.4.2.5.8.3.1

Move $x$ .

$f' (x) = \frac{\frac{2 x^{4} + 4 x^{3} - 8 (x \cdot x^{2}) - 8 x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.3.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.4.2.5.8.3.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{\frac{2 x^{4} + 4 x^{3} - 8 (x \cdot x^{2}) - 8 x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{\frac{2 x^{4} + 4 x^{3} - 8 x^{1 + 2} - 8 x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.3.3

Add $1$ and $2$ .

$f' (x) = \frac{\frac{2 x^{4} + 4 x^{3} - 8 x^{3} - 8 x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.4

Multiply $2$ by $- 8$ .

$f' (x) = \frac{\frac{2 x^{4} + 4 x^{3} - 8 x^{3} - 16 x^{2} + 8 x \cdot x + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.4.2.5.8.5.1

Move $x$ .

$f' (x) = \frac{\frac{2 x^{4} + 4 x^{3} - 8 x^{3} - 16 x^{2} + 8 (x \cdot x) + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.5.2

Multiply $x$ by $x$ .

$f' (x) = \frac{\frac{2 x^{4} + 4 x^{3} - 8 x^{3} - 16 x^{2} + 8 x^{2} + 8 x \cdot 2 - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.8.6

Multiply $2$ by $8$ .

$f' (x) = \frac{\frac{2 x^{4} + 4 x^{3} - 8 x^{3} - 16 x^{2} + 8 x^{2} + 16 x - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.9

Subtract $8 x^{3}$ from $4 x^{3}$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 16 x^{2} + 8 x^{2} + 16 x - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.10

Add $- 16 x^{2}$ and $8 x^{2}$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | (2 + x) (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.11

Expand $(2 + x) (2 - x)$ using the FOIL Method.

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Step 2.1.4.2.5.11.1

Apply the distributive property.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 2 (2 - x) + x (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.11.2

Apply the distributive property.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 2 \cdot 2 + 2 (- x) + x (2 - x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.11.3

Apply the distributive property.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 2 \cdot 2 + 2 (- x) + x \cdot 2 + x (- x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.12

Simplify and combine like terms.

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Step 2.1.4.2.5.12.1

Simplify each term.

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Step 2.1.4.2.5.12.1.1

Multiply $2$ by $2$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 4 + 2 (- x) + x \cdot 2 + x (- x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.12.1.2

Multiply $- 1$ by $2$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 4 - 2 x + x \cdot 2 + x (- x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.12.1.3

Move $2$ to the left of $x$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 4 - 2 x + 2 \cdot x + x (- x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.12.1.4

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 4 - 2 x + 2 x - x \cdot x |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.12.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.4.2.5.12.1.5.1

Move $x$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 4 - 2 x + 2 x - (x \cdot x) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.12.1.5.2

Multiply $x$ by $x$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 4 - 2 x + 2 x - x^{2} |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.12.2

Add $- 2 x$ and $2 x$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 4 + 0 - x^{2} |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.12.3

Add $4$ and $0$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 - x^{2} | | 4 - x^{2} |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.13

Multiply $- | 4 - x^{2} | | 4 - x^{2} |$ .

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Step 2.1.4.2.5.13.1

To multiply absolute values, multiply the terms inside each absolute value.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | (4 - x^{2}) (4 - x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.13.2

Raise $4 - x^{2}$ to the power of $1$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | (4 - x^{2}) (4 - x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.13.3

Raise $4 - x^{2}$ to the power of $1$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | (4 - x^{2}) (4 - x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.13.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | {(4 - x^{2})}^{1 + 1} |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.13.5

Add $1$ and $1$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | {(4 - x^{2})}^{2} |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.14

Rewrite ${(4 - x^{2})}^{2}$ as $(4 - x^{2}) (4 - x^{2})$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | (4 - x^{2}) (4 - x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.15

Expand $(4 - x^{2}) (4 - x^{2})$ using the FOIL Method.

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Step 2.1.4.2.5.15.1

Apply the distributive property.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 (4 - x^{2}) - x^{2} (4 - x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.15.2

Apply the distributive property.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 \cdot 4 + 4 (- x^{2}) - x^{2} (4 - x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.15.3

Apply the distributive property.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 4 \cdot 4 + 4 (- x^{2}) - x^{2} \cdot 4 - x^{2} (- x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.16

Simplify and combine like terms.

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Step 2.1.4.2.5.16.1

Simplify each term.

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Step 2.1.4.2.5.16.1.1

Multiply $4$ by $4$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 + 4 (- x^{2}) - x^{2} \cdot 4 - x^{2} (- x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.16.1.2

Multiply $- 1$ by $4$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 4 x^{2} - x^{2} \cdot 4 - x^{2} (- x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.16.1.3

Multiply $4$ by $- 1$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 4 x^{2} - 4 x^{2} - x^{2} (- x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.16.1.4

Rewrite using the commutative property of multiplication.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 4 x^{2} - 4 x^{2} - 1 \cdot (- 1 x^{2} x^{2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.16.1.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.1.4.2.5.16.1.5.1

Move $x^{2}$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 4 x^{2} - 4 x^{2} - 1 \cdot (- 1 (x^{2} x^{2})) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.16.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 4 x^{2} - 4 x^{2} - 1 \cdot (- 1 x^{2 + 2}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.16.1.5.3

Add $2$ and $2$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 4 x^{2} - 4 x^{2} - 1 \cdot (- 1 x^{4}) |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.16.1.6

Multiply $- 1$ by $- 1$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 4 x^{2} - 4 x^{2} + 1 x^{4} |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.16.1.7

Multiply $x^{4}$ by $1$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 4 x^{2} - 4 x^{2} + x^{4} |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.2.5.16.2

Subtract $4 x^{2}$ from $- 4 x^{2}$ .

$f' (x) = \frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$

Step 2.1.4.3

Combine terms.

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Step 2.1.4.3.1

Rewrite $\frac{\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{| (2 + x) (2 - x) |}}{{(x - 2)}^{2}}$ as a product.

$f' (x) = \frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{| (2 + x) (2 - x) |} \cdot \frac{1}{{(x - 2)}^{2}}$

Step 2.1.4.3.2

Multiply $\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{| (2 + x) (2 - x) |}$ by $\frac{1}{{(x - 2)}^{2}}$ .

$f' (x) = \frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{| (2 + x) (2 - x) | {(x - 2)}^{2}}$

Step 2.1.4.4

Reorder terms.

$f' (x) = \frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{{(x - 2)}^{2} | (2 + x) (2 - x) |}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{{(x - 2)}^{2} | (2 + x) (2 - x) |}$ .

$\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{{(x - 2)}^{2} | (2 + x) (2 - x) |}$

Step 3

Set the first derivative equal to $0$ then solve the equation $\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{{(x - 2)}^{2} | (2 + x) (2 - x) |} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{{(x - 2)}^{2} | (2 + x) (2 - x) |} = 0$

Step 3.2

Set the numerator equal to zero.

$2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} | = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

Move all terms not containing $| 16 - 8 x^{2} + x^{4} |$ to the right side of the equation.

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Step 3.3.1.1

Subtract $2 x^{4}$ from both sides of the equation.

$- 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} | = - 2 x^{4}$

Step 3.3.1.2

Add $4 x^{3}$ to both sides of the equation.

$- 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} | = - 2 x^{4} + 4 x^{3}$

Step 3.3.1.3

Add $8 x^{2}$ to both sides of the equation.

$16 x - | 16 - 8 x^{2} + x^{4} | = - 2 x^{4} + 4 x^{3} + 8 x^{2}$

Step 3.3.1.4

Subtract $16 x$ from both sides of the equation.

$- | 16 - 8 x^{2} + x^{4} | = - 2 x^{4} + 4 x^{3} + 8 x^{2} - 16 x$

Step 3.3.2

Divide each term in $- | 16 - 8 x^{2} + x^{4} | = - 2 x^{4} + 4 x^{3} + 8 x^{2} - 16 x$ by $- 1$ and simplify.

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Step 3.3.2.1

Divide each term in $- | 16 - 8 x^{2} + x^{4} | = - 2 x^{4} + 4 x^{3} + 8 x^{2} - 16 x$ by $- 1$ .

$\frac{- | 16 - 8 x^{2} + x^{4} |}{- 1} = \frac{- 2 x^{4}}{- 1} + \frac{4 x^{3}}{- 1} + \frac{8 x^{2}}{- 1} + \frac{- 16 x}{- 1}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{| 16 - 8 x^{2} + x^{4} |}{1} = \frac{- 2 x^{4}}{- 1} + \frac{4 x^{3}}{- 1} + \frac{8 x^{2}}{- 1} + \frac{- 16 x}{- 1}$

Step 3.3.2.2.2

Divide $| 16 - 8 x^{2} + x^{4} |$ by $1$ .

$| 16 - 8 x^{2} + x^{4} | = \frac{- 2 x^{4}}{- 1} + \frac{4 x^{3}}{- 1} + \frac{8 x^{2}}{- 1} + \frac{- 16 x}{- 1}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Simplify each term.

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Step 3.3.2.3.1.1

Move the negative one from the denominator of $\frac{- 2 x^{4}}{- 1}$ .

$| 16 - 8 x^{2} + x^{4} | = - 1 \cdot (- 2 x^{4}) + \frac{4 x^{3}}{- 1} + \frac{8 x^{2}}{- 1} + \frac{- 16 x}{- 1}$

Step 3.3.2.3.1.2

Rewrite $- 1 \cdot (- 2 x^{4})$ as $- (- 2 x^{4})$ .

$| 16 - 8 x^{2} + x^{4} | = - (- 2 x^{4}) + \frac{4 x^{3}}{- 1} + \frac{8 x^{2}}{- 1} + \frac{- 16 x}{- 1}$

Step 3.3.2.3.1.3

Multiply $- 2$ by $- 1$ .

$| 16 - 8 x^{2} + x^{4} | = 2 x^{4} + \frac{4 x^{3}}{- 1} + \frac{8 x^{2}}{- 1} + \frac{- 16 x}{- 1}$

Step 3.3.2.3.1.4

Move the negative one from the denominator of $\frac{4 x^{3}}{- 1}$ .

$| 16 - 8 x^{2} + x^{4} | = 2 x^{4} - 1 \cdot (4 x^{3}) + \frac{8 x^{2}}{- 1} + \frac{- 16 x}{- 1}$

Step 3.3.2.3.1.5

Rewrite $- 1 \cdot (4 x^{3})$ as $- (4 x^{3})$ .

$| 16 - 8 x^{2} + x^{4} | = 2 x^{4} - (4 x^{3}) + \frac{8 x^{2}}{- 1} + \frac{- 16 x}{- 1}$

Step 3.3.2.3.1.6

Multiply $4$ by $- 1$ .

$| 16 - 8 x^{2} + x^{4} | = 2 x^{4} - 4 x^{3} + \frac{8 x^{2}}{- 1} + \frac{- 16 x}{- 1}$

Step 3.3.2.3.1.7

Move the negative one from the denominator of $\frac{8 x^{2}}{- 1}$ .

$| 16 - 8 x^{2} + x^{4} | = 2 x^{4} - 4 x^{3} - 1 \cdot (8 x^{2}) + \frac{- 16 x}{- 1}$

Step 3.3.2.3.1.8

Rewrite $- 1 \cdot (8 x^{2})$ as $- (8 x^{2})$ .

$| 16 - 8 x^{2} + x^{4} | = 2 x^{4} - 4 x^{3} - (8 x^{2}) + \frac{- 16 x}{- 1}$

Step 3.3.2.3.1.9

Multiply $8$ by $- 1$ .

$| 16 - 8 x^{2} + x^{4} | = 2 x^{4} - 4 x^{3} - 8 x^{2} + \frac{- 16 x}{- 1}$

Step 3.3.2.3.1.10

Move the negative one from the denominator of $\frac{- 16 x}{- 1}$ .

$| 16 - 8 x^{2} + x^{4} | = 2 x^{4} - 4 x^{3} - 8 x^{2} - 1 \cdot (- 16 x)$

Step 3.3.2.3.1.11

Rewrite $- 1 \cdot (- 16 x)$ as $- (- 16 x)$ .

$| 16 - 8 x^{2} + x^{4} | = 2 x^{4} - 4 x^{3} - 8 x^{2} - (- 16 x)$

Step 3.3.2.3.1.12

Multiply $- 16$ by $- 1$ .

$| 16 - 8 x^{2} + x^{4} | = 2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x$

Step 3.3.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$16 - 8 x^{2} + x^{4} = \pm (2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x)$

Step 3.3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$16 - 8 x^{2} + x^{4} = 2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x$

Step 3.3.4.2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x = 16 - 8 x^{2} + x^{4}$

Step 3.3.4.3

Move all terms containing $x$ to the left side of the equation.

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Step 3.3.4.3.1

Add $8 x^{2}$ to both sides of the equation.

$2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x + 8 x^{2} = 16 + x^{4}$

Step 3.3.4.3.2

Subtract $x^{4}$ from both sides of the equation.

$2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x + 8 x^{2} - x^{4} = 16$

Step 3.3.4.3.3

Combine the opposite terms in $2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x + 8 x^{2} - x^{4}$ .

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Step 3.3.4.3.3.1

Add $- 8 x^{2}$ and $8 x^{2}$ .

$2 x^{4} - 4 x^{3} + 16 x + 0 - x^{4} = 16$

Step 3.3.4.3.3.2

Add $2 x^{4} - 4 x^{3} + 16 x$ and $0$ .

$2 x^{4} - 4 x^{3} + 16 x - x^{4} = 16$

Step 3.3.4.3.4

Subtract $x^{4}$ from $2 x^{4}$ .

$x^{4} - 4 x^{3} + 16 x = 16$

Step 3.3.4.4

Subtract $16$ from both sides of the equation.

$x^{4} - 4 x^{3} + 16 x - 16 = 0$

Step 3.3.4.5

Factor the left side of the equation.

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Step 3.3.4.5.1

Factor $x^{4} - 4 x^{3} + 16 x - 16$ using the rational roots test.

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Step 3.3.4.5.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

$q = \pm 1$

Step 3.3.4.5.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

Step 3.3.4.5.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 3.3.4.5.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{4} - 4 {(- 2)}^{3} + 16 \cdot - 2 - 16$

Step 3.3.4.5.1.3.2

Raise $- 2$ to the power of $4$ .

$16 - 4 {(- 2)}^{3} + 16 \cdot - 2 - 16$

Step 3.3.4.5.1.3.3

Raise $- 2$ to the power of $3$ .

$16 - 4 \cdot - 8 + 16 \cdot - 2 - 16$

Step 3.3.4.5.1.3.4

Multiply $- 4$ by $- 8$ .

$16 + 32 + 16 \cdot - 2 - 16$

Step 3.3.4.5.1.3.5

Add $16$ and $32$ .

$48 + 16 \cdot - 2 - 16$

Step 3.3.4.5.1.3.6

Multiply $16$ by $- 2$ .

$48 - 32 - 16$

Step 3.3.4.5.1.3.7

Subtract $32$ from $48$ .

$16 - 16$

Step 3.3.4.5.1.3.8

Subtract $16$ from $16$ .

$0$

Step 3.3.4.5.1.4

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 4 x^{3} + 16 x - 16}{x + 2}$

Step 3.3.4.5.1.5

Divide $x^{4} - 4 x^{3} + 16 x - 16$ by $x + 2$ .

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Step 3.3.4.5.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

Step 3.3.4.5.1.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

Step 3.3.4.5.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

Step 3.3.4.5.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 2 x^{3}$

$x^{3}$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

Step 3.3.4.5.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

Step 3.3.4.5.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

Step 3.3.4.5.1.5.7

Divide the highest order term in the dividend $- 6 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

Step 3.3.4.5.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

Step 3.3.4.5.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{3} - 12 x^{2}$

$x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

Step 3.3.4.5.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

Step 3.3.4.5.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$6 x^{2}$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

$16 x$

Step 3.3.4.5.1.5.12

Divide the highest order term in the dividend $12 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$6 x^{2}$

$12 x$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

$16 x$

Step 3.3.4.5.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$6 x^{2}$

$12 x$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

$16 x$

$12 x^{2}$

$24 x$

Step 3.3.4.5.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{2} + 24 x$

$x^{3}$

$6 x^{2}$

$12 x$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

$16 x$

$12 x^{2}$

$24 x$

Step 3.3.4.5.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$6 x^{2}$

$12 x$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

$16 x$

$12 x^{2}$

$24 x$

$8 x$

Step 3.3.4.5.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$6 x^{2}$

$12 x$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

$16 x$

$12 x^{2}$

$24 x$

$8 x$

$16$

Step 3.3.4.5.1.5.17

Divide the highest order term in the dividend $- 8 x$ by the highest order term in divisor $x$ .

$x^{3}$

$6 x^{2}$

$12 x$

$8$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

$16 x$

$12 x^{2}$

$24 x$

$8 x$

$16$

Step 3.3.4.5.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$6 x^{2}$

$12 x$

$8$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

$16 x$

$12 x^{2}$

$24 x$

$8 x$

$16$

$8 x$

$16$

Step 3.3.4.5.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x - 16$

$x^{3}$

$6 x^{2}$

$12 x$

$8$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

$16 x$

$12 x^{2}$

$24 x$

$8 x$

$16$

$8 x$

$16$

Step 3.3.4.5.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$6 x^{2}$

$12 x$

$8$

$x$

$2$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$6 x^{3}$

$0 x^{2}$

$6 x^{3}$

$12 x^{2}$

$16 x$

$12 x^{2}$

$24 x$

$8 x$

$16$

$8 x$

$16$

$0$

Step 3.3.4.5.1.5.21

Since the remander is $0$ , the final answer is the quotient.

$x^{3} - 6 x^{2} + 12 x - 8$

Step 3.3.4.5.1.6

Write $x^{4} - 4 x^{3} + 16 x - 16$ as a set of factors.

$(x + 2) (x^{3} - 6 x^{2} + 12 x - 8) = 0$

Step 3.3.4.5.2

Factor $x^{3} - 6 x^{2} + 12 x - 8$ using the rational roots test.

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Step 3.3.4.5.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 8, \pm 2, \pm 4$

$q = \pm 1$

Step 3.3.4.5.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 8, \pm 2, \pm 4$

Step 3.3.4.5.2.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 3.3.4.5.2.3.1

Substitute $2$ into the polynomial.

$2^{3} - 6 \cdot 2^{2} + 12 \cdot 2 - 8$

Step 3.3.4.5.2.3.2

Raise $2$ to the power of $3$ .

$8 - 6 \cdot 2^{2} + 12 \cdot 2 - 8$

Step 3.3.4.5.2.3.3

Raise $2$ to the power of $2$ .

$8 - 6 \cdot 4 + 12 \cdot 2 - 8$

Step 3.3.4.5.2.3.4

Multiply $- 6$ by $4$ .

$8 - 24 + 12 \cdot 2 - 8$

Step 3.3.4.5.2.3.5

Subtract $24$ from $8$ .

$- 16 + 12 \cdot 2 - 8$

Step 3.3.4.5.2.3.6

Multiply $12$ by $2$ .

$- 16 + 24 - 8$

Step 3.3.4.5.2.3.7

Add $- 16$ and $24$ .

$8 - 8$

Step 3.3.4.5.2.3.8

Subtract $8$ from $8$ .

$0$

Step 3.3.4.5.2.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 6 x^{2} + 12 x - 8}{x - 2}$

Step 3.3.4.5.2.5

Divide $x^{3} - 6 x^{2} + 12 x - 8$ by $x - 2$ .

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Step 3.3.4.5.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$x^{3}$

$6 x^{2}$

$12 x$

$8$

Step 3.3.4.5.2.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$

Step 3.3.4.5.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			+	$x^{3}$	-	$2 x^{2}$

Step 3.3.4.5.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 2 x^{2}$

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$

Step 3.3.4.5.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$

Step 3.3.4.5.2.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$

Step 3.3.4.5.2.5.7

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$

Step 3.3.4.5.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					-	$4 x^{2}$	+	$8 x$

Step 3.3.4.5.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 8 x$

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$

Step 3.3.4.5.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$

Step 3.3.4.5.2.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$

Step 3.3.4.5.2.5.12

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$

Step 3.3.4.5.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$
							+	$4 x$	-	$8$

Step 3.3.4.5.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x - 8$

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$
							-	$4 x$	+	$8$

Step 3.3.4.5.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$4 x$	+	$4$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$12 x$	-	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$12 x$
					+	$4 x^{2}$	-	$8 x$
							+	$4 x$	-	$8$
							-	$4 x$	+	$8$
										$0$

Step 3.3.4.5.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 4 x + 4$

Step 3.3.4.5.2.6

Write $x^{3} - 6 x^{2} + 12 x - 8$ as a set of factors.

$(x + 2) ((x - 2) (x^{2} - 4 x + 4)) = 0$

Step 3.3.4.5.3

Factor using the perfect square rule.

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Step 3.3.4.5.3.1

Rewrite $4$ as $2^{2}$ .

$(x + 2) ((x - 2) (x^{2} - 4 x + 2^{2})) = 0$

Step 3.3.4.5.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 3.3.4.5.3.3

Rewrite the polynomial.

$(x + 2) ((x - 2) (x^{2} - 2 \cdot x \cdot 2 + 2^{2})) = 0$

Step 3.3.4.5.3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

$(x + 2) ((x - 2) {(x - 2)}^{2}) = 0$

Step 3.3.4.5.4

Combine like factors.

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Step 3.3.4.5.4.1

Raise $x - 2$ to the power of $1$ .

$(x + 2) ((x - 2) {(x - 2)}^{2}) = 0$

Step 3.3.4.5.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 2) {(x - 2)}^{1 + 2} = 0$

Step 3.3.4.5.4.3

Add $1$ and $2$ .

$(x + 2) {(x - 2)}^{3} = 0$

Step 3.3.4.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

${(x - 2)}^{3} = 0$

Step 3.3.4.7

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 3.3.4.7.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.3.4.7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.3.4.8

Set ${(x - 2)}^{3}$ equal to $0$ and solve for $x$ .

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Step 3.3.4.8.1

Set ${(x - 2)}^{3}$ equal to $0$ .

${(x - 2)}^{3} = 0$

Step 3.3.4.8.2

Solve ${(x - 2)}^{3} = 0$ for $x$ .

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Step 3.3.4.8.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.3.4.8.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.3.4.9

The final solution is all the values that make $(x + 2) {(x - 2)}^{3} = 0$ true.

$x = - 2, 2$

Step 3.3.4.10

Next, use the negative value of the $\pm$ to find the second solution.

$16 - 8 x^{2} + x^{4} = - (2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x)$

Step 3.3.4.11

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- (2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x) = 16 - 8 x^{2} + x^{4}$

Step 3.3.4.12

Simplify $- (2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x)$ .

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Step 3.3.4.12.1

Rewrite.

$0 + 0 - (2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x) = 16 - 8 x^{2} + x^{4}$

Step 3.3.4.12.2

Simplify by adding zeros.

$- (2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x) = 16 - 8 x^{2} + x^{4}$

Step 3.3.4.12.3

Apply the distributive property.

$- (2 x^{4}) - (- 4 x^{3}) - (- 8 x^{2}) - (16 x) = 16 - 8 x^{2} + x^{4}$

Step 3.3.4.12.4

Simplify.

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Step 3.3.4.12.4.1

Multiply $2$ by $- 1$ .

$- 2 x^{4} - (- 4 x^{3}) - (- 8 x^{2}) - (16 x) = 16 - 8 x^{2} + x^{4}$

Step 3.3.4.12.4.2

Multiply $- 4$ by $- 1$ .

$- 2 x^{4} + 4 x^{3} - (- 8 x^{2}) - (16 x) = 16 - 8 x^{2} + x^{4}$

Step 3.3.4.12.4.3

Multiply $- 8$ by $- 1$ .

$- 2 x^{4} + 4 x^{3} + 8 x^{2} - (16 x) = 16 - 8 x^{2} + x^{4}$

Step 3.3.4.12.4.4

Multiply $16$ by $- 1$ .

$- 2 x^{4} + 4 x^{3} + 8 x^{2} - 16 x = 16 - 8 x^{2} + x^{4}$

Step 3.3.4.13

Move all terms containing $x$ to the left side of the equation.

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Step 3.3.4.13.1

Add $8 x^{2}$ to both sides of the equation.

$- 2 x^{4} + 4 x^{3} + 8 x^{2} - 16 x + 8 x^{2} = 16 + x^{4}$

Step 3.3.4.13.2

Subtract $x^{4}$ from both sides of the equation.

$- 2 x^{4} + 4 x^{3} + 8 x^{2} - 16 x + 8 x^{2} - x^{4} = 16$

Step 3.3.4.13.3

Subtract $x^{4}$ from $- 2 x^{4}$ .

$- 3 x^{4} + 4 x^{3} + 8 x^{2} - 16 x + 8 x^{2} = 16$

Step 3.3.4.13.4

Add $8 x^{2}$ and $8 x^{2}$ .

$- 3 x^{4} + 4 x^{3} + 16 x^{2} - 16 x = 16$

Step 3.3.4.14

Subtract $16$ from both sides of the equation.

$- 3 x^{4} + 4 x^{3} + 16 x^{2} - 16 x - 16 = 0$

Step 3.3.4.15

Factor the left side of the equation.

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Step 3.3.4.15.1

Regroup terms.

$4 x^{3} - 16 x - 3 x^{4} + 16 x^{2} - 16 = 0$

Step 3.3.4.15.2

Factor $4 x$ out of $4 x^{3} - 16 x$ .

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Step 3.3.4.15.2.1

Factor $4 x$ out of $4 x^{3}$ .

$4 x (x^{2}) - 16 x - 3 x^{4} + 16 x^{2} - 16 = 0$

Step 3.3.4.15.2.2

Factor $4 x$ out of $- 16 x$ .

$4 x (x^{2}) + 4 x (- 4) - 3 x^{4} + 16 x^{2} - 16 = 0$

Step 3.3.4.15.2.3

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 4)$ .

$4 x (x^{2} - 4) - 3 x^{4} + 16 x^{2} - 16 = 0$

Step 3.3.4.15.3

Rewrite $4$ as $2^{2}$ .

$4 x (x^{2} - 2^{2}) - 3 x^{4} + 16 x^{2} - 16 = 0$

Step 3.3.4.15.4

Factor.

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Step 3.3.4.15.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$4 x ((x + 2) (x - 2)) - 3 x^{4} + 16 x^{2} - 16 = 0$

Step 3.3.4.15.4.2

Remove unnecessary parentheses.

$4 x (x + 2) (x - 2) - 3 x^{4} + 16 x^{2} - 16 = 0$

Step 3.3.4.15.5

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$4 x (x + 2) (x - 2) - 3 {(x^{2})}^{2} + 16 x^{2} - 16 = 0$

Step 3.3.4.15.6

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$4 x (x + 2) (x - 2) - 3 u^{2} + 16 u - 16 = 0$

Step 3.3.4.15.7

Factor by grouping.

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Step 3.3.4.15.7.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot - 16 = 48$ and whose sum is $b = 16$ .

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Step 3.3.4.15.7.1.1

Factor $16$ out of $16 u$ .

$4 x (x + 2) (x - 2) - 3 u^{2} + 16 (u) - 16 = 0$

Step 3.3.4.15.7.1.2

Rewrite $16$ as $4$ plus $12$

$4 x (x + 2) (x - 2) - 3 u^{2} + (4 + 12) u - 16 = 0$

Step 3.3.4.15.7.1.3

Apply the distributive property.

$4 x (x + 2) (x - 2) - 3 u^{2} + 4 u + 12 u - 16 = 0$

Step 3.3.4.15.7.2

Factor out the greatest common factor from each group.

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Step 3.3.4.15.7.2.1

Group the first two terms and the last two terms.

$4 x (x + 2) (x - 2) - 3 u^{2} + 4 u + 12 u - 16 = 0$

Step 3.3.4.15.7.2.2

Factor out the greatest common factor (GCF) from each group.

$4 x (x + 2) (x - 2) + u (- 3 u + 4) - 4 (- 3 u + 4) = 0$

Step 3.3.4.15.7.3

Factor the polynomial by factoring out the greatest common factor, $- 3 u + 4$ .

$4 x (x + 2) (x - 2) + (- 3 u + 4) (u - 4) = 0$

Step 3.3.4.15.8

Replace all occurrences of $u$ with $x^{2}$ .

$4 x (x + 2) (x - 2) + (- 3 x^{2} + 4) (x^{2} - 4) = 0$

Step 3.3.4.15.9

Rewrite $4$ as $2^{2}$ .

$4 x (x + 2) (x - 2) + (- 3 x^{2} + 4) (x^{2} - 2^{2}) = 0$

Step 3.3.4.15.10

Factor.

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Step 3.3.4.15.10.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$4 x (x + 2) (x - 2) + (- 3 x^{2} + 4) ((x + 2) (x - 2)) = 0$

Step 3.3.4.15.10.2

Remove unnecessary parentheses.

$4 x (x + 2) (x - 2) + (- 3 x^{2} + 4) (x + 2) (x - 2) = 0$

Step 3.3.4.15.11

Factor $(x + 2) (x - 2)$ out of $4 x (x + 2) (x - 2) + (- 3 x^{2} + 4) (x + 2) (x - 2)$ .

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Step 3.3.4.15.11.1

Factor $(x + 2) (x - 2)$ out of $4 x (x + 2) (x - 2)$ .

$(x + 2) (x - 2) (4 x) + (- 3 x^{2} + 4) (x + 2) (x - 2) = 0$

Step 3.3.4.15.11.2

Factor $(x + 2) (x - 2)$ out of $(- 3 x^{2} + 4) (x + 2) (x - 2)$ .

$(x + 2) (x - 2) (4 x) + (x + 2) (x - 2) (- 3 x^{2} + 4) = 0$

Step 3.3.4.15.11.3

Factor $(x + 2) (x - 2)$ out of $(x + 2) (x - 2) (4 x) + (x + 2) (x - 2) (- 3 x^{2} + 4)$ .

$(x + 2) (x - 2) (4 x - 3 x^{2} + 4) = 0$

Step 3.3.4.15.12

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x + 2) (x - 2) (4 u - 3 u^{2} + 4) = 0$

Step 3.3.4.15.13

Factor by grouping.

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Step 3.3.4.15.13.1

Reorder terms.

$(x + 2) (x - 2) (- 3 u^{2} + 4 u + 4) = 0$

Step 3.3.4.15.13.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot 4 = - 12$ and whose sum is $b = 4$ .

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Step 3.3.4.15.13.2.1

Factor $4$ out of $4 u$ .

$(x + 2) (x - 2) (- 3 u^{2} + 4 (u) + 4) = 0$

Step 3.3.4.15.13.2.2

Rewrite $4$ as $- 2$ plus $6$

$(x + 2) (x - 2) (- 3 u^{2} + (- 2 + 6) u + 4) = 0$

Step 3.3.4.15.13.2.3

Apply the distributive property.

$(x + 2) (x - 2) (- 3 u^{2} - 2 u + 6 u + 4) = 0$

Step 3.3.4.15.13.3

Factor out the greatest common factor from each group.

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Step 3.3.4.15.13.3.1

Group the first two terms and the last two terms.

$(x + 2) (x - 2) ((- 3 u^{2} - 2 u) + 6 u + 4) = 0$

Step 3.3.4.15.13.3.2

Factor out the greatest common factor (GCF) from each group.

$(x + 2) (x - 2) (u (- 3 u - 2) - 2 (- 3 u - 2)) = 0$

Step 3.3.4.15.13.4

Factor the polynomial by factoring out the greatest common factor, $- 3 u - 2$ .

$(x + 2) (x - 2) ((- 3 u - 2) (u - 2)) = 0$

Step 3.3.4.15.14

Factor.

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Step 3.3.4.15.14.1

Replace all occurrences of $u$ with $x$ .

$(x + 2) (x - 2) ((- 3 x - 2) (x - 2)) = 0$

Step 3.3.4.15.14.2

Remove unnecessary parentheses.

$(x + 2) (x - 2) (- 3 x - 2) (x - 2) = 0$

Step 3.3.4.15.15

Combine exponents.

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Step 3.3.4.15.15.1

Raise $x - 2$ to the power of $1$ .

$(x + 2) ((x - 2) (x - 2)) (- 3 x - 2) = 0$

Step 3.3.4.15.15.2

Raise $x - 2$ to the power of $1$ .

$(x + 2) ((x - 2) (x - 2)) (- 3 x - 2) = 0$

Step 3.3.4.15.15.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 2) {(x - 2)}^{1 + 1} (- 3 x - 2) = 0$

Step 3.3.4.15.15.4

Add $1$ and $1$ .

$(x + 2) {(x - 2)}^{2} (- 3 x - 2) = 0$

Step 3.3.4.16

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

${(x - 2)}^{2} = 0$

$- 3 x - 2 = 0$

Step 3.3.4.17

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 3.3.4.17.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.3.4.17.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.3.4.18

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.3.4.18.1

Set ${(x - 2)}^{2}$ equal to $0$ .

${(x - 2)}^{2} = 0$

Step 3.3.4.18.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 3.3.4.18.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.3.4.18.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.3.4.19

Set $- 3 x - 2$ equal to $0$ and solve for $x$ .

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Step 3.3.4.19.1

Set $- 3 x - 2$ equal to $0$ .

$- 3 x - 2 = 0$

Step 3.3.4.19.2

Solve $- 3 x - 2 = 0$ for $x$ .

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Step 3.3.4.19.2.1

Add $2$ to both sides of the equation.

$- 3 x = 2$

Step 3.3.4.19.2.2

Divide each term in $- 3 x = 2$ by $- 3$ and simplify.

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Step 3.3.4.19.2.2.1

Divide each term in $- 3 x = 2$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{2}{- 3}$

Step 3.3.4.19.2.2.2

Simplify the left side.

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Step 3.3.4.19.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.3.4.19.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{2}{- 3}$

Step 3.3.4.19.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{- 3}$

Step 3.3.4.19.2.2.3

Simplify the right side.

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Step 3.3.4.19.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

Step 3.3.4.20

The final solution is all the values that make $(x + 2) {(x - 2)}^{2} (- 3 x - 2) = 0$ true.

$x = - 2, 2, - \frac{2}{3}$

Step 3.4

Exclude the solutions that do not make $\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{{(x - 2)}^{2} | (2 + x) (2 - x) |} = 0$ true.

$No solution$

Step 4

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 5

Find where the derivative is undefined.

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Step 5.1

Set the denominator in $\frac{2 x^{4} - 4 x^{3} - 8 x^{2} + 16 x - | 16 - 8 x^{2} + x^{4} |}{{(x - 2)}^{2} | (2 + x) (2 - x) |}$ equal to $0$ to find where the expression is undefined.

${(x - 2)}^{2} | (2 + x) (2 - x) | = 0$

Step 5.2

Solve for $x$ .

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Step 5.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x - 2)}^{2} = 0$

$| (2 + x) (2 - x) | = 0$

Step 5.2.2

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 5.2.2.1

Set ${(x - 2)}^{2}$ equal to $0$ .

${(x - 2)}^{2} = 0$

Step 5.2.2.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 5.2.2.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 5.2.2.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5.2.3

Set $| (2 + x) (2 - x) |$ equal to $0$ and solve for $x$ .

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Step 5.2.3.1

Set $| (2 + x) (2 - x) |$ equal to $0$ .

$| (2 + x) (2 - x) | = 0$

Step 5.2.3.2

Solve $| (2 + x) (2 - x) | = 0$ for $x$ .

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Step 5.2.3.2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$(2 + x) (2 - x) = \pm 0$

Step 5.2.3.2.2

Plus or minus $0$ is $0$ .

$(2 + x) (2 - x) = 0$

Step 5.2.3.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 + x = 0$

$2 - x = 0$

Step 5.2.3.2.4

Set $2 + x$ equal to $0$ and solve for $x$ .

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Step 5.2.3.2.4.1

Set $2 + x$ equal to $0$ .

$2 + x = 0$

Step 5.2.3.2.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5.2.3.2.5

Set $2 - x$ equal to $0$ and solve for $x$ .

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Step 5.2.3.2.5.1

Set $2 - x$ equal to $0$ .

$2 - x = 0$

Step 5.2.3.2.5.2

Solve $2 - x = 0$ for $x$ .

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Step 5.2.3.2.5.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 5.2.3.2.5.2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 5.2.3.2.5.2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 5.2.3.2.5.2.2.2

Simplify the left side.

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Step 5.2.3.2.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 5.2.3.2.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 5.2.3.2.5.2.2.3

Simplify the right side.

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Step 5.2.3.2.5.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 5.2.3.2.6

The final solution is all the values that make $(2 + x) (2 - x) = 0$ true.

$x = - 2, 2$

Step 5.2.4

The final solution is all the values that make ${(x - 2)}^{2} | (2 + x) (2 - x) | = 0$ true.

$x = 2, - 2$

Step 5.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 2, x = 2$

Step 6

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Step 7

Substitute a value from the interval $(- \infty, - 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 3$ in the expression.

$f' (- 3) = \frac{2 {(- 3)}^{4} - 4 {(- 3)}^{3} - 8 {(- 3)}^{2} + 16 (- 3) - | 16 - 8 {(- 3)}^{2} + {(- 3)}^{4} |}{{((- 3) - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2

Simplify the result.

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Step 7.2.1

Remove parentheses.

$f' (- 3) = \frac{2 {(- 3)}^{4} - 4 {(- 3)}^{3} - 8 {(- 3)}^{2} + 16 (- 3) - | 16 - 8 {(- 3)}^{2} + {(- 3)}^{4} |}{{((- 3) - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Raise $- 3$ to the power of $4$ .

$f' (- 3) = \frac{2 \cdot 81 - 4 {(- 3)}^{3} - 8 {(- 3)}^{2} + 16 (- 3) - | 16 - 8 {(- 3)}^{2} + {(- 3)}^{4} |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.2

Multiply $2$ by $81$ .

$f' (- 3) = \frac{162 - 4 {(- 3)}^{3} - 8 {(- 3)}^{2} + 16 (- 3) - | 16 - 8 {(- 3)}^{2} + {(- 3)}^{4} |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.3

Raise $- 3$ to the power of $3$ .

$f' (- 3) = \frac{162 - 4 \cdot - 27 - 8 {(- 3)}^{2} + 16 (- 3) - | 16 - 8 {(- 3)}^{2} + {(- 3)}^{4} |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.4

Multiply $- 4$ by $- 27$ .

$f' (- 3) = \frac{162 + 108 - 8 {(- 3)}^{2} + 16 (- 3) - | 16 - 8 {(- 3)}^{2} + {(- 3)}^{4} |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.5

Raise $- 3$ to the power of $2$ .

$f' (- 3) = \frac{162 + 108 - 8 \cdot 9 + 16 (- 3) - | 16 - 8 {(- 3)}^{2} + {(- 3)}^{4} |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.6

Multiply $- 8$ by $9$ .

$f' (- 3) = \frac{162 + 108 - 72 + 16 (- 3) - | 16 - 8 {(- 3)}^{2} + {(- 3)}^{4} |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.7

Multiply $16$ by $- 3$ .

$f' (- 3) = \frac{162 + 108 - 72 - 48 - | 16 - 8 {(- 3)}^{2} + {(- 3)}^{4} |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.8

Simplify each term.

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Step 7.2.2.8.1

Raise $- 3$ to the power of $2$ .

$f' (- 3) = \frac{162 + 108 - 72 - 48 - | 16 - 8 \cdot 9 + {(- 3)}^{4} |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.8.2

Multiply $- 8$ by $9$ .

$f' (- 3) = \frac{162 + 108 - 72 - 48 - | 16 - 72 + {(- 3)}^{4} |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.8.3

Raise $- 3$ to the power of $4$ .

$f' (- 3) = \frac{162 + 108 - 72 - 48 - | 16 - 72 + 81 |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.9

Subtract $72$ from $16$ .

$f' (- 3) = \frac{162 + 108 - 72 - 48 - | - 56 + 81 |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.10

Add $- 56$ and $81$ .

$f' (- 3) = \frac{162 + 108 - 72 - 48 - | 25 |}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.11

The absolute value is the distance between a number and zero. The distance between $0$ and $25$ is $25$ .

$f' (- 3) = \frac{162 + 108 - 72 - 48 - 1 \cdot 25}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.12

Multiply $- 1$ by $25$ .

$f' (- 3) = \frac{162 + 108 - 72 - 48 - 25}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.13

Add $162$ and $108$ .

$f' (- 3) = \frac{270 - 72 - 48 - 25}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.14

Subtract $72$ from $270$ .

$f' (- 3) = \frac{198 - 48 - 25}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.15

Subtract $48$ from $198$ .

$f' (- 3) = \frac{150 - 25}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.2.16

Subtract $25$ from $150$ .

$f' (- 3) = \frac{125}{{(- 3 - 2)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Subtract $2$ from $- 3$ .

$f' (- 3) = \frac{125}{{(- 5)}^{2} | (2 - 3) (2 - (- 3)) |}$

Step 7.2.3.2

Subtract $3$ from $2$ .

$f' (- 3) = \frac{125}{{(- 5)}^{2} | - 1 (2 - (- 3)) |}$

Step 7.2.3.3

Multiply $- 1$ by $- 3$ .

$f' (- 3) = \frac{125}{{(- 5)}^{2} | - 1 (2 + 3) |}$

Step 7.2.3.4

Add $2$ and $3$ .

$f' (- 3) = \frac{125}{{(- 5)}^{2} | - 1 \cdot 5 |}$

Step 7.2.3.5

Raise $- 5$ to the power of $2$ .

$f' (- 3) = \frac{125}{25 | - 1 \cdot 5 |}$

Step 7.2.3.6

Multiply $- 1$ by $5$ .

$f' (- 3) = \frac{125}{25 | - 5 |}$

Step 7.2.3.7

The absolute value is the distance between a number and zero. The distance between $- 5$ and $0$ is $5$ .

$f' (- 3) = \frac{125}{25 \cdot 5}$

Step 7.2.4

Simplify the expression.

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Step 7.2.4.1

Multiply $25$ by $5$ .

$f' (- 3) = \frac{125}{125}$

Step 7.2.4.2

Divide $125$ by $125$ .

$f' (- 3) = 1$

Step 7.2.5

The final answer is $1$ .

$1$

Step 7.3

At $x = - 3$ the derivative is $1$ . Since this is positive, the function is increasing on $(- \infty, - 2)$ .

Increasing on $(- \infty, - 2)$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(- 2, 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = \frac{2 {(0)}^{4} - 4 {(0)}^{3} - 8 {(0)}^{2} + 16 (0) - | 16 - 8 {(0)}^{2} + {(0)}^{4} |}{{((0) - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2

Simplify the result.

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Step 8.2.1

Remove parentheses.

$f' (0) = \frac{2 {(0)}^{4} - 4 {(0)}^{3} - 8 {(0)}^{2} + 16 (0) - | 16 - 8 {(0)}^{2} + {(0)}^{4} |}{{((0) - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2

Simplify the numerator.

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Step 8.2.2.1

Raising $0$ to any positive power yields $0$ .

$f' (0) = \frac{2 \cdot 0 - 4 \cdot 0^{3} - 8 \cdot 0^{2} + 16 (0) - | 16 - 8 \cdot 0^{2} + 0^{4} |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.2

Multiply $2$ by $0$ .

$f' (0) = \frac{0 - 4 \cdot 0^{3} - 8 \cdot 0^{2} + 16 (0) - | 16 - 8 \cdot 0^{2} + 0^{4} |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.3

Raising $0$ to any positive power yields $0$ .

$f' (0) = \frac{0 - 4 \cdot 0 - 8 \cdot 0^{2} + 16 (0) - | 16 - 8 \cdot 0^{2} + 0^{4} |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.4

Multiply $- 4$ by $0$ .

$f' (0) = \frac{0 + 0 - 8 \cdot 0^{2} + 16 (0) - | 16 - 8 \cdot 0^{2} + 0^{4} |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.5

Raising $0$ to any positive power yields $0$ .

$f' (0) = \frac{0 + 0 - 8 \cdot 0 + 16 (0) - | 16 - 8 \cdot 0^{2} + 0^{4} |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.6

Multiply $- 8$ by $0$ .

$f' (0) = \frac{0 + 0 + 0 + 16 (0) - | 16 - 8 \cdot 0^{2} + 0^{4} |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.7

Multiply $16$ by $0$ .

$f' (0) = \frac{0 + 0 + 0 + 0 - | 16 - 8 \cdot 0^{2} + 0^{4} |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.8

Simplify each term.

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Step 8.2.2.8.1

Raising $0$ to any positive power yields $0$ .

$f' (0) = \frac{0 + 0 + 0 + 0 - | 16 - 8 \cdot 0 + 0^{4} |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.8.2

Multiply $- 8$ by $0$ .

$f' (0) = \frac{0 + 0 + 0 + 0 - | 16 + 0 + 0^{4} |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.8.3

Raising $0$ to any positive power yields $0$ .

$f' (0) = \frac{0 + 0 + 0 + 0 - | 16 + 0 + 0 |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.9

Add $16$ and $0$ .

$f' (0) = \frac{0 + 0 + 0 + 0 - | 16 + 0 |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.10

Add $16$ and $0$ .

$f' (0) = \frac{0 + 0 + 0 + 0 - | 16 |}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.11

The absolute value is the distance between a number and zero. The distance between $0$ and $16$ is $16$ .

$f' (0) = \frac{0 + 0 + 0 + 0 - 1 \cdot 16}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.12

Multiply $- 1$ by $16$ .

$f' (0) = \frac{0 + 0 + 0 + 0 - 16}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.13

Add $0$ and $0$ .

$f' (0) = \frac{0 + 0 + 0 - 16}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.14

Add $0$ and $0$ .

$f' (0) = \frac{0 + 0 - 16}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.15

Add $0$ and $0$ .

$f' (0) = \frac{0 - 16}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.2.16

Subtract $16$ from $0$ .

$f' (0) = \frac{- 16}{{(0 - 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.3

Simplify the denominator.

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Step 8.2.3.1

Subtract $2$ from $0$ .

$f' (0) = \frac{- 16}{{(- 2)}^{2} | (2 + 0) (2 - (0)) |}$

Step 8.2.3.2

Add $2$ and $0$ .

$f' (0) = \frac{- 16}{{(- 2)}^{2} | 2 (2 - (0)) |}$

Step 8.2.3.3

Multiply $- 1$ by $0$ .

$f' (0) = \frac{- 16}{{(- 2)}^{2} | 2 (2 + 0) |}$

Step 8.2.3.4

Add $2$ and $0$ .

$f' (0) = \frac{- 16}{{(- 2)}^{2} | 2 \cdot 2 |}$

Step 8.2.3.5

Raise $- 2$ to the power of $2$ .

$f' (0) = \frac{- 16}{4 | 2 \cdot 2 |}$

Step 8.2.3.6

Multiply $2$ by $2$ .

$f' (0) = \frac{- 16}{4 | 4 |}$

Step 8.2.3.7

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$f' (0) = \frac{- 16}{4 \cdot 4}$

Step 8.2.4

Simplify the expression.

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Step 8.2.4.1

Multiply $4$ by $4$ .

$f' (0) = \frac{- 16}{16}$

Step 8.2.4.2

Divide $- 16$ by $16$ .

$f' (0) = - 1$

Step 8.2.5

The final answer is $- 1$ .

$- 1$

Step 8.3

At $x = 0$ the derivative is $- 1$ . Since this is negative, the function is decreasing on $(- 2, 2)$ .

Decreasing on $(- 2, 2)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(2, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = \frac{2 {(3)}^{4} - 4 {(3)}^{3} - 8 {(3)}^{2} + 16 (3) - | 16 - 8 {(3)}^{2} + {(3)}^{4} |}{{((3) - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2

Simplify the result.

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Step 9.2.1

Remove parentheses.

$f' (3) = \frac{2 {(3)}^{4} - 4 {(3)}^{3} - 8 {(3)}^{2} + 16 (3) - | 16 - 8 {(3)}^{2} + {(3)}^{4} |}{{((3) - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2

Simplify the numerator.

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Step 9.2.2.1

Raise $3$ to the power of $4$ .

$f' (3) = \frac{2 \cdot 81 - 4 \cdot 3^{3} - 8 \cdot 3^{2} + 16 (3) - | 16 - 8 \cdot 3^{2} + 3^{4} |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.2

Multiply $2$ by $81$ .

$f' (3) = \frac{162 - 4 \cdot 3^{3} - 8 \cdot 3^{2} + 16 (3) - | 16 - 8 \cdot 3^{2} + 3^{4} |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.3

Raise $3$ to the power of $3$ .

$f' (3) = \frac{162 - 4 \cdot 27 - 8 \cdot 3^{2} + 16 (3) - | 16 - 8 \cdot 3^{2} + 3^{4} |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.4

Multiply $- 4$ by $27$ .

$f' (3) = \frac{162 - 108 - 8 \cdot 3^{2} + 16 (3) - | 16 - 8 \cdot 3^{2} + 3^{4} |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.5

Raise $3$ to the power of $2$ .

$f' (3) = \frac{162 - 108 - 8 \cdot 9 + 16 (3) - | 16 - 8 \cdot 3^{2} + 3^{4} |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.6

Multiply $- 8$ by $9$ .

$f' (3) = \frac{162 - 108 - 72 + 16 (3) - | 16 - 8 \cdot 3^{2} + 3^{4} |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.7

Multiply $16$ by $3$ .

$f' (3) = \frac{162 - 108 - 72 + 48 - | 16 - 8 \cdot 3^{2} + 3^{4} |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.8

Simplify each term.

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Step 9.2.2.8.1

Raise $3$ to the power of $2$ .

$f' (3) = \frac{162 - 108 - 72 + 48 - | 16 - 8 \cdot 9 + 3^{4} |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.8.2

Multiply $- 8$ by $9$ .

$f' (3) = \frac{162 - 108 - 72 + 48 - | 16 - 72 + 3^{4} |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.8.3

Raise $3$ to the power of $4$ .

$f' (3) = \frac{162 - 108 - 72 + 48 - | 16 - 72 + 81 |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.9

Subtract $72$ from $16$ .

$f' (3) = \frac{162 - 108 - 72 + 48 - | - 56 + 81 |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.10

Add $- 56$ and $81$ .

$f' (3) = \frac{162 - 108 - 72 + 48 - | 25 |}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.11

The absolute value is the distance between a number and zero. The distance between $0$ and $25$ is $25$ .

$f' (3) = \frac{162 - 108 - 72 + 48 - 1 \cdot 25}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.12

Multiply $- 1$ by $25$ .

$f' (3) = \frac{162 - 108 - 72 + 48 - 25}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.13

Subtract $108$ from $162$ .

$f' (3) = \frac{54 - 72 + 48 - 25}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.14

Subtract $72$ from $54$ .

$f' (3) = \frac{- 18 + 48 - 25}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.15

Add $- 18$ and $48$ .

$f' (3) = \frac{30 - 25}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.2.16

Subtract $25$ from $30$ .

$f' (3) = \frac{5}{{(3 - 2)}^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.3

Simplify the denominator.

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Step 9.2.3.1

Subtract $2$ from $3$ .

$f' (3) = \frac{5}{1^{2} | (2 + 3) (2 - (3)) |}$

Step 9.2.3.2

Add $2$ and $3$ .

$f' (3) = \frac{5}{1^{2} | 5 (2 - (3)) |}$

Step 9.2.3.3

Multiply $- 1$ by $3$ .

$f' (3) = \frac{5}{1^{2} | 5 (2 - 3) |}$

Step 9.2.3.4

Subtract $3$ from $2$ .

$f' (3) = \frac{5}{1^{2} | 5 \cdot - 1 |}$

Step 9.2.3.5

One to any power is one.

$f' (3) = \frac{5}{1 | 5 \cdot - 1 |}$

Step 9.2.3.6

Multiply $5$ by $- 1$ .

$f' (3) = \frac{5}{1 | - 5 |}$

Step 9.2.3.7

The absolute value is the distance between a number and zero. The distance between $- 5$ and $0$ is $5$ .

$f' (3) = \frac{5}{1 \cdot 5}$

Step 9.2.3.8

Multiply $5$ by $1$ .

$f' (3) = \frac{5}{5}$

Step 9.2.4

Divide $5$ by $5$ .

$f' (3) = 1$

Step 9.2.5

The final answer is $1$ .

$1$

Step 9.3

At $x = 3$ the derivative is $1$ . Since this is positive, the function is increasing on $(2, \infty)$ .

Increasing on $(2, \infty)$ since $f' (x) > 0$

Step 10

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, - 2), (2, \infty)$

Decreasing on: $(- 2, 2)$

Step 11