Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{3}$ and $g (x) = {(2 x^{2} + x - 3)}^{2}$ .

$x^{3} \frac{d}{d x} [{(2 x^{2} + x - 3)}^{2}] + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 2 x^{2} + x - 3$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $2 x^{2} + x - 3$ .

$x^{3} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [2 x^{2} + x - 3]) + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.2.2

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

$x^{3} (2 u \frac{d}{d x} [2 x^{2} + x - 3]) + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.2.3

Replace all occurrences of $u$ with $2 x^{2} + x - 3$ .

$x^{3} (2 (2 x^{2} + x - 3) \frac{d}{d x} [2 x^{2} + x - 3]) + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

By the Sum Rule, the derivative of $2 x^{2} + x - 3$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$x^{3} (2 (2 x^{2} + x - 3) (\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 3])) + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.3.2

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

$x^{3} (2 (2 x^{2} + x - 3) (2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 3])) + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.3.3

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

$x^{3} (2 (2 x^{2} + x - 3) (2 (2 x) + \frac{d}{d x} [x] + \frac{d}{d x} [- 3])) + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.3.4

Multiply $2$ by $2$ .

$x^{3} (2 (2 x^{2} + x - 3) (4 x + \frac{d}{d x} [x] + \frac{d}{d x} [- 3])) + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.3.5

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

$x^{3} (2 (2 x^{2} + x - 3) (4 x + 1 + \frac{d}{d x} [- 3])) + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.3.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$x^{3} (2 (2 x^{2} + x - 3) (4 x + 1 + 0)) + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.3.7

Add $4 x + 1$ and $0$ .

$x^{3} (2 (2 x^{2} + x - 3) (4 x + 1)) + {(2 x^{2} + x - 3)}^{2} \frac{d}{d x} [x^{3}]$

Step 1.1.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{3} (2 (2 x^{2} + x - 3) (4 x + 1)) + {(2 x^{2} + x - 3)}^{2} (3 x^{2})$

Step 1.1.3.9

Move $3$ to the left of ${(2 x^{2} + x - 3)}^{2}$ .

$x^{3} (2 (2 x^{2} + x - 3) (4 x + 1)) + 3 \cdot {(2 x^{2} + x - 3)}^{2} x^{2}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Apply the distributive property.

$x^{3} ((2 (2 x^{2}) + 2 x + 2 \cdot - 3) (4 x + 1)) + 3 {(2 x^{2} + x - 3)}^{2} x^{2}$

Step 1.1.4.2

Multiply $2$ by $2$ .

$x^{3} ((4 x^{2} + 2 x + 2 \cdot - 3) (4 x + 1)) + 3 {(2 x^{2} + x - 3)}^{2} x^{2}$

Step 1.1.4.3

Multiply $2$ by $- 3$ .

$x^{3} ((4 x^{2} + 2 x - 6) (4 x + 1)) + 3 {(2 x^{2} + x - 3)}^{2} x^{2}$

Step 1.1.4.4

Factor $x^{2}$ out of $x^{3} ((4 x^{2} + 2 x - 6) (4 x + 1)) + 3 {(2 x^{2} + x - 3)}^{2} x^{2}$ .

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Step 1.1.4.4.1

Factor $x^{2}$ out of $x^{3} ((4 x^{2} + 2 x - 6) (4 x + 1))$ .

$x^{2} (x ((4 x^{2} + 2 x - 6) (4 x + 1))) + 3 {(2 x^{2} + x - 3)}^{2} x^{2}$

Step 1.1.4.4.2

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

$x^{2} (x ((4 x^{2} + 2 x - 6) (4 x + 1))) + x^{2} (3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.4.3

Factor $x^{2}$ out of $x^{2} (x ((4 x^{2} + 2 x - 6) (4 x + 1))) + x^{2} (3 {(2 x^{2} + x - 3)}^{2})$ .

$x^{2} (x ((4 x^{2} + 2 x - 6) (4 x + 1)) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5

Simplify each term.

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Step 1.1.4.5.1

Expand $(4 x^{2} + 2 x - 6) (4 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} (x (4 x^{2} (4 x) + 4 x^{2} \cdot 1 + 2 x (4 x) + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2

Simplify each term.

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Step 1.1.4.5.2.1

Rewrite using the commutative property of multiplication.

$x^{2} (x (4 \cdot 4 x^{2} x + 4 x^{2} \cdot 1 + 2 x (4 x) + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.2

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

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Step 1.1.4.5.2.2.1

Move $x$ .

$x^{2} (x (4 \cdot 4 (x \cdot x^{2}) + 4 x^{2} \cdot 1 + 2 x (4 x) + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.2.2

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

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Step 1.1.4.5.2.2.2.1

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

$x^{2} (x (4 \cdot 4 (x^{1} x^{2}) + 4 x^{2} \cdot 1 + 2 x (4 x) + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.2.2.2

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

$x^{2} (x (4 \cdot 4 x^{1 + 2} + 4 x^{2} \cdot 1 + 2 x (4 x) + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.2.3

Add $1$ and $2$ .

$x^{2} (x (4 \cdot 4 x^{3} + 4 x^{2} \cdot 1 + 2 x (4 x) + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.3

Multiply $4$ by $4$ .

$x^{2} (x (16 x^{3} + 4 x^{2} \cdot 1 + 2 x (4 x) + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.4

Multiply $4$ by $1$ .

$x^{2} (x (16 x^{3} + 4 x^{2} + 2 x (4 x) + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.5

Rewrite using the commutative property of multiplication.

$x^{2} (x (16 x^{3} + 4 x^{2} + 2 \cdot 4 x \cdot x + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.6

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

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Step 1.1.4.5.2.6.1

Move $x$ .

$x^{2} (x (16 x^{3} + 4 x^{2} + 2 \cdot 4 (x \cdot x) + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.6.2

Multiply $x$ by $x$ .

$x^{2} (x (16 x^{3} + 4 x^{2} + 2 \cdot 4 x^{2} + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.7

Multiply $2$ by $4$ .

$x^{2} (x (16 x^{3} + 4 x^{2} + 8 x^{2} + 2 x \cdot 1 - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.8

Multiply $2$ by $1$ .

$x^{2} (x (16 x^{3} + 4 x^{2} + 8 x^{2} + 2 x - 6 (4 x) - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.9

Multiply $4$ by $- 6$ .

$x^{2} (x (16 x^{3} + 4 x^{2} + 8 x^{2} + 2 x - 24 x - 6 \cdot 1) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.2.10

Multiply $- 6$ by $1$ .

$x^{2} (x (16 x^{3} + 4 x^{2} + 8 x^{2} + 2 x - 24 x - 6) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.3

Add $4 x^{2}$ and $8 x^{2}$ .

$x^{2} (x (16 x^{3} + 12 x^{2} + 2 x - 24 x - 6) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.4

Subtract $24 x$ from $2 x$ .

$x^{2} (x (16 x^{3} + 12 x^{2} - 22 x - 6) + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.5

Apply the distributive property.

$x^{2} (x (16 x^{3}) + x (12 x^{2}) + x (- 22 x) + x \cdot - 6 + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.6

Simplify.

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Step 1.1.4.5.6.1

Rewrite using the commutative property of multiplication.

$x^{2} (16 x \cdot x^{3} + x (12 x^{2}) + x (- 22 x) + x \cdot - 6 + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.6.2

Rewrite using the commutative property of multiplication.

$x^{2} (16 x \cdot x^{3} + 12 x \cdot x^{2} + x (- 22 x) + x \cdot - 6 + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.6.3

Rewrite using the commutative property of multiplication.

$x^{2} (16 x \cdot x^{3} + 12 x \cdot x^{2} - 22 x \cdot x + x \cdot - 6 + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.6.4

Move $- 6$ to the left of $x$ .

$x^{2} (16 x \cdot x^{3} + 12 x \cdot x^{2} - 22 x \cdot x - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.7

Simplify each term.

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Step 1.1.4.5.7.1

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

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Step 1.1.4.5.7.1.1

Move $x^{3}$ .

$x^{2} (16 (x^{3} x) + 12 x \cdot x^{2} - 22 x \cdot x - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.7.1.2

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

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Step 1.1.4.5.7.1.2.1

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

$x^{2} (16 (x^{3} x^{1}) + 12 x \cdot x^{2} - 22 x \cdot x - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.7.1.2.2

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

$x^{2} (16 x^{3 + 1} + 12 x \cdot x^{2} - 22 x \cdot x - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.7.1.3

Add $3$ and $1$ .

$x^{2} (16 x^{4} + 12 x \cdot x^{2} - 22 x \cdot x - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.7.2

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

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Step 1.1.4.5.7.2.1

Move $x^{2}$ .

$x^{2} (16 x^{4} + 12 (x^{2} x) - 22 x \cdot x - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.7.2.2

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

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Step 1.1.4.5.7.2.2.1

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

$x^{2} (16 x^{4} + 12 (x^{2} x^{1}) - 22 x \cdot x - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.7.2.2.2

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

$x^{2} (16 x^{4} + 12 x^{2 + 1} - 22 x \cdot x - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.7.2.3

Add $2$ and $1$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x \cdot x - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.7.3

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

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Step 1.1.4.5.7.3.1

Move $x$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 (x \cdot x) - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.7.3.2

Multiply $x$ by $x$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 \cdot x + 3 {(2 x^{2} + x - 3)}^{2})$

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 {(2 x^{2} + x - 3)}^{2})$

Step 1.1.4.5.8

Rewrite ${(2 x^{2} + x - 3)}^{2}$ as $(2 x^{2} + x - 3) (2 x^{2} + x - 3)$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 ((2 x^{2} + x - 3) (2 x^{2} + x - 3)))$

Step 1.1.4.5.9

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

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (2 x^{2} (2 x^{2}) + 2 x^{2} x + 2 x^{2} \cdot - 3 + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10

Simplify each term.

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Step 1.1.4.5.10.1

Rewrite using the commutative property of multiplication.

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (2 \cdot 2 x^{2} x^{2} + 2 x^{2} x + 2 x^{2} \cdot - 3 + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.2

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

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Step 1.1.4.5.10.2.1

Move $x^{2}$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (2 \cdot 2 (x^{2} x^{2}) + 2 x^{2} x + 2 x^{2} \cdot - 3 + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.2.2

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

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (2 \cdot 2 x^{2 + 2} + 2 x^{2} x + 2 x^{2} \cdot - 3 + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.2.3

Add $2$ and $2$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (2 \cdot 2 x^{4} + 2 x^{2} x + 2 x^{2} \cdot - 3 + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.3

Multiply $2$ by $2$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{2} x + 2 x^{2} \cdot - 3 + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.4

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

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Step 1.1.4.5.10.4.1

Move $x$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 (x \cdot x^{2}) + 2 x^{2} \cdot - 3 + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.4.2

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

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Step 1.1.4.5.10.4.2.1

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

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 (x^{1} x^{2}) + 2 x^{2} \cdot - 3 + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.4.2.2

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

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{1 + 2} + 2 x^{2} \cdot - 3 + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.4.3

Add $1$ and $2$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} + 2 x^{2} \cdot - 3 + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.5

Multiply $- 3$ by $2$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} - 6 x^{2} + x (2 x^{2}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.6

Rewrite using the commutative property of multiplication.

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} - 6 x^{2} + 2 x \cdot x^{2} + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.7

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

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Step 1.1.4.5.10.7.1

Move $x^{2}$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} - 6 x^{2} + 2 (x^{2} x) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.7.2

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

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Step 1.1.4.5.10.7.2.1

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

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} - 6 x^{2} + 2 (x^{2} x^{1}) + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.7.2.2

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

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} - 6 x^{2} + 2 x^{2 + 1} + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.7.3

Add $2$ and $1$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} - 6 x^{2} + 2 x^{3} + x \cdot x + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.8

Multiply $x$ by $x$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} - 6 x^{2} + 2 x^{3} + x^{2} + x \cdot - 3 - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.9

Move $- 3$ to the left of $x$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} - 6 x^{2} + 2 x^{3} + x^{2} - 3 \cdot x - 3 (2 x^{2}) - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.10

Multiply $2$ by $- 3$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} - 6 x^{2} + 2 x^{3} + x^{2} - 3 x - 6 x^{2} - 3 x - 3 \cdot - 3))$

Step 1.1.4.5.10.11

Multiply $- 3$ by $- 3$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 2 x^{3} - 6 x^{2} + 2 x^{3} + x^{2} - 3 x - 6 x^{2} - 3 x + 9))$

Step 1.1.4.5.11

Add $2 x^{3}$ and $2 x^{3}$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 4 x^{3} - 6 x^{2} + x^{2} - 3 x - 6 x^{2} - 3 x + 9))$

Step 1.1.4.5.12

Add $- 6 x^{2}$ and $x^{2}$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 4 x^{3} - 5 x^{2} - 3 x - 6 x^{2} - 3 x + 9))$

Step 1.1.4.5.13

Subtract $6 x^{2}$ from $- 5 x^{2}$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 4 x^{3} - 11 x^{2} - 3 x - 3 x + 9))$

Step 1.1.4.5.14

Subtract $3 x$ from $- 3 x$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4} + 4 x^{3} - 11 x^{2} - 6 x + 9))$

Step 1.1.4.5.15

Apply the distributive property.

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 3 (4 x^{4}) + 3 (4 x^{3}) + 3 (- 11 x^{2}) + 3 (- 6 x) + 3 \cdot 9)$

Step 1.1.4.5.16

Simplify.

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Step 1.1.4.5.16.1

Multiply $4$ by $3$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 12 x^{4} + 3 (4 x^{3}) + 3 (- 11 x^{2}) + 3 (- 6 x) + 3 \cdot 9)$

Step 1.1.4.5.16.2

Multiply $4$ by $3$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 12 x^{4} + 12 x^{3} + 3 (- 11 x^{2}) + 3 (- 6 x) + 3 \cdot 9)$

Step 1.1.4.5.16.3

Multiply $- 11$ by $3$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 12 x^{4} + 12 x^{3} - 33 x^{2} + 3 (- 6 x) + 3 \cdot 9)$

Step 1.1.4.5.16.4

Multiply $- 6$ by $3$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 12 x^{4} + 12 x^{3} - 33 x^{2} - 18 x + 3 \cdot 9)$

Step 1.1.4.5.16.5

Multiply $3$ by $9$ .

$x^{2} (16 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 12 x^{4} + 12 x^{3} - 33 x^{2} - 18 x + 27)$

Step 1.1.4.6

Add $16 x^{4}$ and $12 x^{4}$ .

$x^{2} (28 x^{4} + 12 x^{3} - 22 x^{2} - 6 x + 12 x^{3} - 33 x^{2} - 18 x + 27)$

Step 1.1.4.7

Add $12 x^{3}$ and $12 x^{3}$ .

$x^{2} (28 x^{4} + 24 x^{3} - 22 x^{2} - 6 x - 33 x^{2} - 18 x + 27)$

Step 1.1.4.8

Subtract $33 x^{2}$ from $- 22 x^{2}$ .

$x^{2} (28 x^{4} + 24 x^{3} - 55 x^{2} - 6 x - 18 x + 27)$

Step 1.1.4.9

Subtract $18 x$ from $- 6 x$ .

$x^{2} (28 x^{4} + 24 x^{3} - 55 x^{2} - 24 x + 27)$

Step 1.1.4.10

Apply the distributive property.

$x^{2} (28 x^{4}) + x^{2} (24 x^{3}) + x^{2} (- 55 x^{2}) + x^{2} (- 24 x) + x^{2} \cdot 27$

Step 1.1.4.11

Simplify.

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Step 1.1.4.11.1

Rewrite using the commutative property of multiplication.

$28 x^{2} x^{4} + x^{2} (24 x^{3}) + x^{2} (- 55 x^{2}) + x^{2} (- 24 x) + x^{2} \cdot 27$

Step 1.1.4.11.2

Rewrite using the commutative property of multiplication.

$28 x^{2} x^{4} + 24 x^{2} x^{3} + x^{2} (- 55 x^{2}) + x^{2} (- 24 x) + x^{2} \cdot 27$

Step 1.1.4.11.3

Rewrite using the commutative property of multiplication.

$28 x^{2} x^{4} + 24 x^{2} x^{3} - 55 x^{2} x^{2} + x^{2} (- 24 x) + x^{2} \cdot 27$

Step 1.1.4.11.4

Rewrite using the commutative property of multiplication.

$28 x^{2} x^{4} + 24 x^{2} x^{3} - 55 x^{2} x^{2} - 24 x^{2} x + x^{2} \cdot 27$

Step 1.1.4.11.5

Move $27$ to the left of $x^{2}$ .

$28 x^{2} x^{4} + 24 x^{2} x^{3} - 55 x^{2} x^{2} - 24 x^{2} x + 27 \cdot x^{2}$

Step 1.1.4.12

Simplify each term.

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Step 1.1.4.12.1

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 1.1.4.12.1.1

Move $x^{4}$ .

$28 (x^{4} x^{2}) + 24 x^{2} x^{3} - 55 x^{2} x^{2} - 24 x^{2} x + 27 \cdot x^{2}$

Step 1.1.4.12.1.2

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

$28 x^{4 + 2} + 24 x^{2} x^{3} - 55 x^{2} x^{2} - 24 x^{2} x + 27 \cdot x^{2}$

Step 1.1.4.12.1.3

Add $4$ and $2$ .

$28 x^{6} + 24 x^{2} x^{3} - 55 x^{2} x^{2} - 24 x^{2} x + 27 \cdot x^{2}$

Step 1.1.4.12.2

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.1.4.12.2.1

Move $x^{3}$ .

$28 x^{6} + 24 (x^{3} x^{2}) - 55 x^{2} x^{2} - 24 x^{2} x + 27 \cdot x^{2}$

Step 1.1.4.12.2.2

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

$28 x^{6} + 24 x^{3 + 2} - 55 x^{2} x^{2} - 24 x^{2} x + 27 \cdot x^{2}$

Step 1.1.4.12.2.3

Add $3$ and $2$ .

$28 x^{6} + 24 x^{5} - 55 x^{2} x^{2} - 24 x^{2} x + 27 \cdot x^{2}$

Step 1.1.4.12.3

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

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Step 1.1.4.12.3.1

Move $x^{2}$ .

$28 x^{6} + 24 x^{5} - 55 (x^{2} x^{2}) - 24 x^{2} x + 27 \cdot x^{2}$

Step 1.1.4.12.3.2

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

$28 x^{6} + 24 x^{5} - 55 x^{2 + 2} - 24 x^{2} x + 27 \cdot x^{2}$

Step 1.1.4.12.3.3

Add $2$ and $2$ .

$28 x^{6} + 24 x^{5} - 55 x^{4} - 24 x^{2} x + 27 \cdot x^{2}$

Step 1.1.4.12.4

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

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Step 1.1.4.12.4.1

Move $x$ .

$28 x^{6} + 24 x^{5} - 55 x^{4} - 24 (x \cdot x^{2}) + 27 \cdot x^{2}$

Step 1.1.4.12.4.2

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

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Step 1.1.4.12.4.2.1

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

$28 x^{6} + 24 x^{5} - 55 x^{4} - 24 (x^{1} x^{2}) + 27 \cdot x^{2}$

Step 1.1.4.12.4.2.2

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

$28 x^{6} + 24 x^{5} - 55 x^{4} - 24 x^{1 + 2} + 27 \cdot x^{2}$

Step 1.1.4.12.4.3

Add $1$ and $2$ .

$f' (x) = 28 x^{6} + 24 x^{5} - 55 x^{4} - 24 x^{3} + 27 \cdot x^{2}$

$f' (x) = 28 x^{6} + 24 x^{5} - 55 x^{4} - 24 x^{3} + 27 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $28 x^{6} + 24 x^{5} - 55 x^{4} - 24 x^{3} + 27 x^{2}$ .

$28 x^{6} + 24 x^{5} - 55 x^{4} - 24 x^{3} + 27 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $28 x^{6} + 24 x^{5} - 55 x^{4} - 24 x^{3} + 27 x^{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$28 x^{6} + 24 x^{5} - 55 x^{4} - 24 x^{3} + 27 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $x^{2}$ out of $28 x^{6} + 24 x^{5} - 55 x^{4} - 24 x^{3} + 27 x^{2}$ .

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Step 2.2.1.1

Factor $x^{2}$ out of $28 x^{6}$ .

$x^{2} (28 x^{4}) + 24 x^{5} - 55 x^{4} - 24 x^{3} + 27 x^{2} = 0$

Step 2.2.1.2

Factor $x^{2}$ out of $24 x^{5}$ .

$x^{2} (28 x^{4}) + x^{2} (24 x^{3}) - 55 x^{4} - 24 x^{3} + 27 x^{2} = 0$

Step 2.2.1.3

Factor $x^{2}$ out of $- 55 x^{4}$ .

$x^{2} (28 x^{4}) + x^{2} (24 x^{3}) + x^{2} (- 55 x^{2}) - 24 x^{3} + 27 x^{2} = 0$

Step 2.2.1.4

Factor $x^{2}$ out of $- 24 x^{3}$ .

$x^{2} (28 x^{4}) + x^{2} (24 x^{3}) + x^{2} (- 55 x^{2}) + x^{2} (- 24 x) + 27 x^{2} = 0$

Step 2.2.1.5

Factor $x^{2}$ out of $27 x^{2}$ .

$x^{2} (28 x^{4}) + x^{2} (24 x^{3}) + x^{2} (- 55 x^{2}) + x^{2} (- 24 x) + x^{2} \cdot 27 = 0$

Step 2.2.1.6

Factor $x^{2}$ out of $x^{2} (28 x^{4}) + x^{2} (24 x^{3})$ .

$x^{2} (28 x^{4} + 24 x^{3}) + x^{2} (- 55 x^{2}) + x^{2} (- 24 x) + x^{2} \cdot 27 = 0$

Step 2.2.1.7

Factor $x^{2}$ out of $x^{2} (28 x^{4} + 24 x^{3}) + x^{2} (- 55 x^{2})$ .

$x^{2} (28 x^{4} + 24 x^{3} - 55 x^{2}) + x^{2} (- 24 x) + x^{2} \cdot 27 = 0$

Step 2.2.1.8

Factor $x^{2}$ out of $x^{2} (28 x^{4} + 24 x^{3} - 55 x^{2}) + x^{2} (- 24 x)$ .

$x^{2} (28 x^{4} + 24 x^{3} - 55 x^{2} - 24 x) + x^{2} \cdot 27 = 0$

Step 2.2.1.9

Factor $x^{2}$ out of $x^{2} (28 x^{4} + 24 x^{3} - 55 x^{2} - 24 x) + x^{2} \cdot 27$ .

$x^{2} (28 x^{4} + 24 x^{3} - 55 x^{2} - 24 x + 27) = 0$

Step 2.2.2

Regroup terms.

$x^{2} (24 x^{3} - 24 x + 28 x^{4} - 55 x^{2} + 27) = 0$

Step 2.2.3

Factor $24 x$ out of $24 x^{3} - 24 x$ .

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Step 2.2.3.1

Factor $24 x$ out of $24 x^{3}$ .

$x^{2} (24 x (x^{2}) - 24 x + 28 x^{4} - 55 x^{2} + 27) = 0$

Step 2.2.3.2

Factor $24 x$ out of $- 24 x$ .

$x^{2} (24 x (x^{2}) + 24 x (- 1) + 28 x^{4} - 55 x^{2} + 27) = 0$

Step 2.2.3.3

Factor $24 x$ out of $24 x (x^{2}) + 24 x (- 1)$ .

$x^{2} (24 x (x^{2} - 1) + 28 x^{4} - 55 x^{2} + 27) = 0$

Step 2.2.4

Rewrite $1$ as $1^{2}$ .

$x^{2} (24 x (x^{2} - 1^{2}) + 28 x^{4} - 55 x^{2} + 27) = 0$

Step 2.2.5

Factor.

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Step 2.2.5.1

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 $b = 1$ .

$x^{2} (24 x ((x + 1) (x - 1)) + 28 x^{4} - 55 x^{2} + 27) = 0$

Step 2.2.5.2

Remove unnecessary parentheses.

$x^{2} (24 x (x + 1) (x - 1) + 28 x^{4} - 55 x^{2} + 27) = 0$

Step 2.2.6

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

$x^{2} (24 x (x + 1) (x - 1) + 28 {(x^{2})}^{2} - 55 x^{2} + 27) = 0$

Step 2.2.7

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

$x^{2} (24 x (x + 1) (x - 1) + 28 u^{2} - 55 u + 27) = 0$

Step 2.2.8

Factor by grouping.

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Step 2.2.8.1

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 $a \cdot c = 28 \cdot 27 = 756$ and whose sum is $b = - 55$ .

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Step 2.2.8.1.1

Factor $- 55$ out of $- 55 u$ .

$x^{2} (24 x (x + 1) (x - 1) + 28 u^{2} - 55 u + 27) = 0$

Step 2.2.8.1.2

Rewrite $- 55$ as $- 27$ plus $- 28$

$x^{2} (24 x (x + 1) (x - 1) + 28 u^{2} + (- 27 - 28) u + 27) = 0$

Step 2.2.8.1.3

Apply the distributive property.

$x^{2} (24 x (x + 1) (x - 1) + 28 u^{2} - 27 u - 28 u + 27) = 0$

Step 2.2.8.2

Factor out the greatest common factor from each group.

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Step 2.2.8.2.1

Group the first two terms and the last two terms.

$x^{2} (24 x (x + 1) (x - 1) + 28 u^{2} - 27 u - 28 u + 27) = 0$

Step 2.2.8.2.2

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

$x^{2} (24 x (x + 1) (x - 1) + u (28 u - 27) - (28 u - 27)) = 0$

Step 2.2.8.3

Factor the polynomial by factoring out the greatest common factor, $28 u - 27$ .

$x^{2} (24 x (x + 1) (x - 1) + (28 u - 27) (u - 1)) = 0$

Step 2.2.9

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

$x^{2} (24 x (x + 1) (x - 1) + (28 x^{2} - 27) (x^{2} - 1)) = 0$

Step 2.2.10

Rewrite $1$ as $1^{2}$ .

$x^{2} (24 x (x + 1) (x - 1) + (28 x^{2} - 27) (x^{2} - 1^{2})) = 0$

Step 2.2.11

Factor.

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Step 2.2.11.1

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 $b = 1$ .

$x^{2} (24 x (x + 1) (x - 1) + (28 x^{2} - 27) ((x + 1) (x - 1))) = 0$

Step 2.2.11.2

Remove unnecessary parentheses.

$x^{2} (24 x (x + 1) (x - 1) + (28 x^{2} - 27) (x + 1) (x - 1)) = 0$

Step 2.2.12

Factor $(x + 1) (x - 1)$ out of $24 x (x + 1) (x - 1) + (28 x^{2} - 27) (x + 1) (x - 1)$ .

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Step 2.2.12.1

Factor $(x + 1) (x - 1)$ out of $24 x (x + 1) (x - 1)$ .

$x^{2} ((x + 1) (x - 1) (24 x) + (28 x^{2} - 27) (x + 1) (x - 1)) = 0$

Step 2.2.12.2

Factor $(x + 1) (x - 1)$ out of $(28 x^{2} - 27) (x + 1) (x - 1)$ .

$x^{2} ((x + 1) (x - 1) (24 x) + (x + 1) (x - 1) (28 x^{2} - 27)) = 0$

Step 2.2.12.3

Factor $(x + 1) (x - 1)$ out of $(x + 1) (x - 1) (24 x) + (x + 1) (x - 1) (28 x^{2} - 27)$ .

$x^{2} ((x + 1) (x - 1) (24 x + 28 x^{2} - 27)) = 0$

Step 2.2.13

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

$x^{2} ((x + 1) (x - 1) (24 u + 28 u^{2} - 27)) = 0$

Step 2.2.14

Factor by grouping.

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Step 2.2.14.1

Reorder terms.

$x^{2} ((x + 1) (x - 1) (28 u^{2} + 24 u - 27)) = 0$

Step 2.2.14.2

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 $a \cdot c = 28 \cdot - 27 = - 756$ and whose sum is $b = 24$ .

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Step 2.2.14.2.1

Factor $24$ out of $24 u$ .

$x^{2} ((x + 1) (x - 1) (28 u^{2} + 24 (u) - 27)) = 0$

Step 2.2.14.2.2

Rewrite $24$ as $- 18$ plus $42$

$x^{2} ((x + 1) (x - 1) (28 u^{2} + (- 18 + 42) u - 27)) = 0$

Step 2.2.14.2.3

Apply the distributive property.

$x^{2} ((x + 1) (x - 1) (28 u^{2} - 18 u + 42 u - 27)) = 0$

Step 2.2.14.3

Factor out the greatest common factor from each group.

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Step 2.2.14.3.1

Group the first two terms and the last two terms.

$x^{2} ((x + 1) (x - 1) ((28 u^{2} - 18 u) + 42 u - 27)) = 0$

Step 2.2.14.3.2

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

$x^{2} ((x + 1) (x - 1) (2 u (14 u - 9) + 3 (14 u - 9))) = 0$

Step 2.2.14.4

Factor the polynomial by factoring out the greatest common factor, $14 u - 9$ .

$x^{2} ((x + 1) (x - 1) ((14 u - 9) (2 u + 3))) = 0$

Step 2.2.15

Factor.

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Step 2.2.15.1

Factor.

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Step 2.2.15.1.1

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

$x^{2} ((x + 1) (x - 1) ((14 x - 9) (2 x + 3))) = 0$

Step 2.2.15.1.2

Remove unnecessary parentheses.

$x^{2} ((x + 1) (x - 1) (14 x - 9) (2 x + 3)) = 0$

Step 2.2.15.2

Remove unnecessary parentheses.

$x^{2} (x + 1) (x - 1) (14 x - 9) (2 x + 3) = 0$

Step 2.3

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

$x^{2} = 0$

$x + 1 = 0$

$x - 1 = 0$

$14 x - 9 = 0$

$2 x + 3 = 0$

Step 2.4

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

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Step 2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.6

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.7

Set $14 x - 9$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $14 x - 9$ equal to $0$ .

$14 x - 9 = 0$

Step 2.7.2

Solve $14 x - 9 = 0$ for $x$ .

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Step 2.7.2.1

Add $9$ to both sides of the equation.

$14 x = 9$

Step 2.7.2.2

Divide each term in $14 x = 9$ by $14$ and simplify.

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Step 2.7.2.2.1

Divide each term in $14 x = 9$ by $14$ .

$\frac{14 x}{14} = \frac{9}{14}$

Step 2.7.2.2.2

Simplify the left side.

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Step 2.7.2.2.2.1

Cancel the common factor of $14$ .

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Step 2.7.2.2.2.1.1

Cancel the common factor.

$\frac{14 x}{14} = \frac{9}{14}$

Step 2.7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{14}$

Step 2.8

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

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Step 2.8.1

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

$2 x + 3 = 0$

Step 2.8.2

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

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Step 2.8.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 2.8.2.2

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

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Step 2.8.2.2.1

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

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

Step 2.8.2.2.2

Simplify the left side.

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Step 2.8.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.8.2.2.2.1.1

Cancel the common factor.

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

Step 2.8.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.8.2.2.3

Simplify the right side.

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Step 2.8.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.9

The final solution is all the values that make $x^{2} (x + 1) (x - 1) (14 x - 9) (2 x + 3) = 0$ true.

$x = 0, - 1, 1, \frac{9}{14}, - \frac{3}{2}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $x^{3} {(2 x^{2} + x - 3)}^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

Remove parentheses.

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

Step 4.1.2.1.2

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

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

Step 4.1.2.2

Simplify each term.

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Step 4.1.2.2.1

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

$0 {(2 \cdot 0 + 0 - 3)}^{2}$

Step 4.1.2.2.2

Multiply $2$ by $0$ .

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

Step 4.1.2.3

Simplify the expression.

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Step 4.1.2.3.1

Add $0$ and $0$ .

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

Step 4.1.2.3.2

Subtract $3$ from $0$ .

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

Step 4.1.2.3.3

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

$0 \cdot 9$

Step 4.1.2.3.4

Multiply $0$ by $9$ .

$0$

Step 4.2

Evaluate at $x = - 1$ .

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Step 4.2.1

Substitute $- 1$ for $x$ .

${(- 1)}^{3} {(2 {(- 1)}^{2} - 1 - 3)}^{2}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the expression.

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Step 4.2.2.1.1

Remove parentheses.

${(- 1)}^{3} {(2 {(- 1)}^{2} - 1 - 3)}^{2}$

Step 4.2.2.1.2

Raise $- 1$ to the power of $3$ .

$- 1 {(2 {(- 1)}^{2} - 1 - 3)}^{2}$

Step 4.2.2.2

Simplify each term.

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Step 4.2.2.2.1

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

$- 1 {(2 \cdot 1 - 1 - 3)}^{2}$

Step 4.2.2.2.2

Multiply $2$ by $1$ .

$- 1 {(2 - 1 - 3)}^{2}$

Step 4.2.2.3

Simplify the expression.

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Step 4.2.2.3.1

Subtract $1$ from $2$ .

$- 1 {(1 - 3)}^{2}$

Step 4.2.2.3.2

Subtract $3$ from $1$ .

$- 1 {(- 2)}^{2}$

Step 4.2.2.3.3

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

$- 1 \cdot 4$

Step 4.2.2.3.4

Multiply $- 1$ by $4$ .

$- 4$

Step 4.3

Evaluate at $x = 1$ .

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Step 4.3.1

Substitute $1$ for $x$ .

${(1)}^{3} {(2 {(1)}^{2} + 1 - 3)}^{2}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the expression.

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Step 4.3.2.1.1

Remove parentheses.

${(1)}^{3} {(2 {(1)}^{2} + 1 - 3)}^{2}$

Step 4.3.2.1.2

One to any power is one.

$1 {(2 {(1)}^{2} + 1 - 3)}^{2}$

Step 4.3.2.1.3

Multiply ${(2 {(1)}^{2} + 1 - 3)}^{2}$ by $1$ .

${(2 {(1)}^{2} + 1 - 3)}^{2}$

Step 4.3.2.2

Simplify each term.

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Step 4.3.2.2.1

One to any power is one.

${(2 \cdot 1 + 1 - 3)}^{2}$

Step 4.3.2.2.2

Multiply $2$ by $1$ .

${(2 + 1 - 3)}^{2}$

Step 4.3.2.3

Simplify the expression.

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Step 4.3.2.3.1

Add $2$ and $1$ .

${(3 - 3)}^{2}$

Step 4.3.2.3.2

Subtract $3$ from $3$ .

$0^{2}$

Step 4.3.2.3.3

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

$0$

Step 4.4

Evaluate at $x = \frac{9}{14}$ .

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Step 4.4.1

Substitute $\frac{9}{14}$ for $x$ .

${(\frac{9}{14})}^{3} {(2 {(\frac{9}{14})}^{2} + \frac{9}{14} - 3)}^{2}$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Simplify the expression.

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Step 4.4.2.1.1

Remove parentheses.

${(\frac{9}{14})}^{3} {(2 {(\frac{9}{14})}^{2} + \frac{9}{14} - 3)}^{2}$

Step 4.4.2.1.2

Apply the product rule to $\frac{9}{14}$ .

$\frac{9^{3}}{14^{3}} {(2 {(\frac{9}{14})}^{2} + \frac{9}{14} - 3)}^{2}$

Step 4.4.2.1.3

Raise $9$ to the power of $3$ .

$\frac{729}{14^{3}} {(2 {(\frac{9}{14})}^{2} + \frac{9}{14} - 3)}^{2}$

Step 4.4.2.1.4

Raise $14$ to the power of $3$ .

$\frac{729}{2744} {(2 {(\frac{9}{14})}^{2} + \frac{9}{14} - 3)}^{2}$

Step 4.4.2.2

Simplify each term.

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Step 4.4.2.2.1

Apply the product rule to $\frac{9}{14}$ .

$\frac{729}{2744} {(2 \frac{9^{2}}{14^{2}} + \frac{9}{14} - 3)}^{2}$

Step 4.4.2.2.2

Raise $9$ to the power of $2$ .

$\frac{729}{2744} {(2 \frac{81}{14^{2}} + \frac{9}{14} - 3)}^{2}$

Step 4.4.2.2.3

Raise $14$ to the power of $2$ .

$\frac{729}{2744} {(2 (\frac{81}{196}) + \frac{9}{14} - 3)}^{2}$

Step 4.4.2.2.4

Cancel the common factor of $2$ .

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Step 4.4.2.2.4.1

Factor $2$ out of $196$ .

$\frac{729}{2744} {(2 \frac{81}{2 (98)} + \frac{9}{14} - 3)}^{2}$

Step 4.4.2.2.4.2

Cancel the common factor.

$\frac{729}{2744} {(2 \frac{81}{2 \cdot 98} + \frac{9}{14} - 3)}^{2}$

Step 4.4.2.2.4.3

Rewrite the expression.

$\frac{729}{2744} {(\frac{81}{98} + \frac{9}{14} - 3)}^{2}$

Step 4.4.2.3

Find the common denominator.

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Step 4.4.2.3.1

Multiply $\frac{9}{14}$ by $\frac{7}{7}$ .

$\frac{729}{2744} {(\frac{81}{98} + \frac{9}{14} \cdot \frac{7}{7} - 3)}^{2}$

Step 4.4.2.3.2

Multiply $\frac{9}{14}$ by $\frac{7}{7}$ .

$\frac{729}{2744} {(\frac{81}{98} + \frac{9 \cdot 7}{14 \cdot 7} - 3)}^{2}$

Step 4.4.2.3.3

Write $- 3$ as a fraction with denominator $1$ .

$\frac{729}{2744} {(\frac{81}{98} + \frac{9 \cdot 7}{14 \cdot 7} + \frac{- 3}{1})}^{2}$

Step 4.4.2.3.4

Multiply $\frac{- 3}{1}$ by $\frac{98}{98}$ .

$\frac{729}{2744} {(\frac{81}{98} + \frac{9 \cdot 7}{14 \cdot 7} + \frac{- 3}{1} \cdot \frac{98}{98})}^{2}$

Step 4.4.2.3.5

Multiply $\frac{- 3}{1}$ by $\frac{98}{98}$ .

$\frac{729}{2744} {(\frac{81}{98} + \frac{9 \cdot 7}{14 \cdot 7} + \frac{- 3 \cdot 98}{98})}^{2}$

Step 4.4.2.3.6

Reorder the factors of $14 \cdot 7$ .

$\frac{729}{2744} {(\frac{81}{98} + \frac{9 \cdot 7}{7 \cdot 14} + \frac{- 3 \cdot 98}{98})}^{2}$

Step 4.4.2.3.7

Multiply $7$ by $14$ .

$\frac{729}{2744} {(\frac{81}{98} + \frac{9 \cdot 7}{98} + \frac{- 3 \cdot 98}{98})}^{2}$

Step 4.4.2.4

Combine the numerators over the common denominator.

$\frac{729}{2744} {(\frac{81 + 9 \cdot 7 - 3 \cdot 98}{98})}^{2}$

Step 4.4.2.5

Simplify each term.

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Step 4.4.2.5.1

Multiply $9$ by $7$ .

$\frac{729}{2744} {(\frac{81 + 63 - 3 \cdot 98}{98})}^{2}$

Step 4.4.2.5.2

Multiply $- 3$ by $98$ .

$\frac{729}{2744} {(\frac{81 + 63 - 294}{98})}^{2}$

Step 4.4.2.6

Reduce the expression by cancelling the common factors.

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Step 4.4.2.6.1

Add $81$ and $63$ .

$\frac{729}{2744} {(\frac{144 - 294}{98})}^{2}$

Step 4.4.2.6.2

Subtract $294$ from $144$ .

$\frac{729}{2744} {(\frac{- 150}{98})}^{2}$

Step 4.4.2.6.3

Cancel the common factor of $- 150$ and $98$ .

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Step 4.4.2.6.3.1

Factor $2$ out of $- 150$ .

$\frac{729}{2744} {(\frac{2 (- 75)}{98})}^{2}$

Step 4.4.2.6.3.2

Cancel the common factors.

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Step 4.4.2.6.3.2.1

Factor $2$ out of $98$ .

$\frac{729}{2744} {(\frac{2 \cdot - 75}{2 \cdot 49})}^{2}$

Step 4.4.2.6.3.2.2

Cancel the common factor.

$\frac{729}{2744} {(\frac{2 \cdot - 75}{2 \cdot 49})}^{2}$

Step 4.4.2.6.3.2.3

Rewrite the expression.

$\frac{729}{2744} {(\frac{- 75}{49})}^{2}$

Step 4.4.2.6.4

Move the negative in front of the fraction.

$\frac{729}{2744} {(- \frac{75}{49})}^{2}$

Step 4.4.2.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.4.2.7.1

Apply the product rule to $- \frac{75}{49}$ .

$\frac{729}{2744} ({(- 1)}^{2} {(\frac{75}{49})}^{2})$

Step 4.4.2.7.2

Apply the product rule to $\frac{75}{49}$ .

$\frac{729}{2744} ({(- 1)}^{2} \frac{75^{2}}{49^{2}})$

Step 4.4.2.8

Simplify the expression.

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Step 4.4.2.8.1

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

$\frac{729}{2744} (1 \frac{75^{2}}{49^{2}})$

Step 4.4.2.8.2

Multiply $\frac{75^{2}}{49^{2}}$ by $1$ .

$\frac{729}{2744} \cdot \frac{75^{2}}{49^{2}}$

Step 4.4.2.9

Combine.

$\frac{729 \cdot 75^{2}}{2744 \cdot 49^{2}}$

Step 4.4.2.10

Simplify the expression.

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Step 4.4.2.10.1

Raise $75$ to the power of $2$ .

$\frac{729 \cdot 5625}{2744 \cdot 49^{2}}$

Step 4.4.2.10.2

Raise $49$ to the power of $2$ .

$\frac{729 \cdot 5625}{2744 \cdot 2401}$

Step 4.4.2.10.3

Multiply $729$ by $5625$ .

$\frac{4100625}{2744 \cdot 2401}$

Step 4.4.2.10.4

Multiply $2744$ by $2401$ .

$\frac{4100625}{6588344}$

Step 4.5

Evaluate at $x = - \frac{3}{2}$ .

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Step 4.5.1

Substitute $- \frac{3}{2}$ for $x$ .

${(- \frac{3}{2})}^{3} {(2 {(- \frac{3}{2})}^{2} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2

Simplify.

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Step 4.5.2.1

Remove parentheses.

${(- \frac{3}{2})}^{3} {(2 {(- \frac{3}{2})}^{2} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.5.2.2.1

Apply the product rule to $- \frac{3}{2}$ .

${(- 1)}^{3} {(\frac{3}{2})}^{3} {(2 {(- \frac{3}{2})}^{2} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.2.2

Apply the product rule to $\frac{3}{2}$ .

${(- 1)}^{3} \frac{3^{3}}{2^{3}} {(2 {(- \frac{3}{2})}^{2} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.3

Evaluate the exponents.

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Step 4.5.2.3.1

Raise $- 1$ to the power of $3$ .

$- \frac{3^{3}}{2^{3}} {(2 {(- \frac{3}{2})}^{2} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.3.2

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

$- \frac{27}{2^{3}} {(2 {(- \frac{3}{2})}^{2} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.3.3

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

$- \frac{27}{8} {(2 {(- \frac{3}{2})}^{2} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.4

Simplify each term.

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Step 4.5.2.4.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.5.2.4.1.1

Apply the product rule to $- \frac{3}{2}$ .

$- \frac{27}{8} {(2 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.4.1.2

Apply the product rule to $\frac{3}{2}$ .

$- \frac{27}{8} {(2 ({(- 1)}^{2} \frac{3^{2}}{2^{2}}) - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.4.2

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

$- \frac{27}{8} {(2 (1 \frac{3^{2}}{2^{2}}) - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.4.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$- \frac{27}{8} {(2 \frac{3^{2}}{2^{2}} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.4.4

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

$- \frac{27}{8} {(2 \frac{9}{2^{2}} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.4.5

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

$- \frac{27}{8} {(2 (\frac{9}{4}) - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.4.6

Cancel the common factor of $2$ .

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Step 4.5.2.4.6.1

Factor $2$ out of $4$ .

$- \frac{27}{8} {(2 \frac{9}{2 (2)} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.4.6.2

Cancel the common factor.

$- \frac{27}{8} {(2 \frac{9}{2 \cdot 2} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.4.6.3

Rewrite the expression.

$- \frac{27}{8} {(\frac{9}{2} - \frac{3}{2} - 3)}^{2}$

Step 4.5.2.5

Combine fractions.

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Step 4.5.2.5.1

Combine the numerators over the common denominator.

$- \frac{27}{8} {(- 3 + \frac{9 - 3}{2})}^{2}$

Step 4.5.2.5.2

Simplify the expression.

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Step 4.5.2.5.2.1

Subtract $3$ from $9$ .

$- \frac{27}{8} {(- 3 + \frac{6}{2})}^{2}$

Step 4.5.2.5.2.2

Divide $6$ by $2$ .

$- \frac{27}{8} {(- 3 + 3)}^{2}$

Step 4.5.2.5.2.3

Add $- 3$ and $3$ .

$- \frac{27}{8} \cdot 0^{2}$

Step 4.5.2.5.2.4

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

$- \frac{27}{8} \cdot 0$

Step 4.5.2.6

Multiply $- \frac{27}{8} \cdot 0$ .

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Step 4.5.2.6.1

Multiply $0$ by $- 1$ .

$0 (\frac{27}{8})$

Step 4.5.2.6.2

Multiply $0$ by $\frac{27}{8}$ .

$0$

Step 4.6

List all of the points.

$(0, 0), (- 1, - 4), (1, 0), (\frac{9}{14}, \frac{4100625}{6588344}), (- \frac{3}{2}, 0)$

Step 5