Algebra Examples

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

Differentiate using the Quotient Rule which states that $\frac{d}{d \cdot d} [\frac{f (d)}{g (d)}]$ is $\frac{g (d) \frac{d}{d \cdot d} [f (d)] - f (d) \frac{d}{d \cdot d} [g (d)]}{{g (d)}^{2}}$ where $f (d) = (x^{2} - 3 x^{2} + 2 x - 1) d x$ and $g (d) = x^{2} - 4 x + 4$ .

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

Differentiate.

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Subtract $3 x^{2}$ from $x^{2}$ .

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

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

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

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

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

Multiply $- 2 x^{2} + 2 x - 1$ by $1$ .

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

Simplify.

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Apply the distributive property.

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

Combine terms.

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

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

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

Since $x^{2}$ is constant with respect to $d$ , the derivative of $x^{2}$ with respect to $d$ is $0$ .

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

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

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

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

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

Combine terms.

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Add $0$ and $0$ .

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

Add $0$ and $0$ .

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

Simplify.

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify the numerator.

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Simplify each term.

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Expand $(x^{2} - 4 x + 4) (- 2 x^{3} + 2 x^{2} - x)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify each term.

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Rewrite using the commutative property of multiplication.

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

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

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Move $x^{3}$ .

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

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

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

Add $3$ and $2$ .

$\frac{- 2 x^{5} + x^{2} (2 x^{2}) + x^{2} (- x) - 4 x (- 2 x^{3}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{- 2 x^{5} + 2 x^{2} x^{2} + x^{2} (- x) - 4 x (- 2 x^{3}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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Move $x^{2}$ .

$\frac{- 2 x^{5} + 2 (x^{2} x^{2}) + x^{2} (- x) - 4 x (- 2 x^{3}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

$\frac{- 2 x^{5} + 2 x^{2 + 2} + x^{2} (- x) - 4 x (- 2 x^{3}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Add $2$ and $2$ .

$\frac{- 2 x^{5} + 2 x^{4} + x^{2} (- x) - 4 x (- 2 x^{3}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{- 2 x^{5} + 2 x^{4} - x^{2} x - 4 x (- 2 x^{3}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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Move $x$ .

$\frac{- 2 x^{5} + 2 x^{4} - (x \cdot x^{2}) - 4 x (- 2 x^{3}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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

$\frac{- 2 x^{5} + 2 x^{4} - (x^{1} x^{2}) - 4 x (- 2 x^{3}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

$\frac{- 2 x^{5} + 2 x^{4} - x^{1 + 2} - 4 x (- 2 x^{3}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Add $1$ and $2$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} - 4 x (- 2 x^{3}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} - 4 \cdot - 2 x \cdot x^{3} - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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Move $x^{3}$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} - 4 \cdot - 2 (x^{3} x) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} - 4 \cdot - 2 (x^{3} x^{1}) - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} - 4 \cdot - 2 x^{3 + 1} - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Add $3$ and $1$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} - 4 \cdot - 2 x^{4} - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $- 4$ by $- 2$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 4 x (2 x^{2}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 4 \cdot 2 x \cdot x^{2} - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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Move $x^{2}$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 4 \cdot 2 (x^{2} x) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 4 \cdot 2 (x^{2} x^{1}) - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 4 \cdot 2 x^{2 + 1} - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Add $2$ and $1$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 4 \cdot 2 x^{3} - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $- 4$ by $2$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 8 x^{3} - 4 x (- x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 8 x^{3} - 4 \cdot - 1 x \cdot x + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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Move $x$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 8 x^{3} - 4 \cdot - 1 (x \cdot x) + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $x$ by $x$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 8 x^{3} - 4 \cdot - 1 x^{2} + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $- 4$ by $- 1$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 8 x^{3} + 4 x^{2} + 4 (- 2 x^{3}) + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $- 2$ by $4$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 8 x^{3} + 4 x^{2} - 8 x^{3} + 4 (2 x^{2}) + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $2$ by $4$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 8 x^{3} + 4 x^{2} - 8 x^{3} + 8 x^{2} + 4 (- x) - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $- 1$ by $4$ .

$\frac{- 2 x^{5} + 2 x^{4} - x^{3} + 8 x^{4} - 8 x^{3} + 4 x^{2} - 8 x^{3} + 8 x^{2} - 4 x - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

$\frac{- 2 x^{5} + 10 x^{4} - x^{3} - 8 x^{3} + 4 x^{2} - 8 x^{3} + 8 x^{2} - 4 x - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

$\frac{- 2 x^{5} + 10 x^{4} - 9 x^{3} + 4 x^{2} - 8 x^{3} + 8 x^{2} - 4 x - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Subtract $8 x^{3}$ from $- 9 x^{3}$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 4 x^{2} + 8 x^{2} - 4 x - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x - x^{2} d x \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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Move $x$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x - (x \cdot x^{2}) d \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x - (x^{1} x^{2}) d \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x - x^{1 + 2} d \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Add $1$ and $2$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x - x^{3} d \cdot 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $- x^{3} d \cdot 0$ .

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Multiply $0$ by $- 1$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 x^{3} d - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $0$ by $x^{3}$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 d - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $0$ by $d$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 - (- 3 x^{2}) d x \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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Move $x$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 - (- 3 (x \cdot x^{2})) d \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 - (- 3 (x^{1} x^{2})) d \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 - (- 3 x^{1 + 2}) d \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Add $1$ and $2$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 - (- 3 x^{3}) d \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $- 3$ by $- 1$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 3 x^{3} d \cdot 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $3 x^{3} d \cdot 0$ .

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Multiply $0$ by $3$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 x^{3} d - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $0$ by $x^{3}$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 d - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $0$ by $d$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 - (2 x) d x \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

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

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Move $x$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 - (2 (x \cdot x)) d \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $x$ by $x$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 - (2 x^{2}) d \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $2$ by $- 1$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 - 2 x^{2} d \cdot 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $- 2 x^{2} d \cdot 0$ .

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Multiply $0$ by $- 2$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 + 0 x^{2} d - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $0$ by $x^{2}$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 + 0 d - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $0$ by $d$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 + 0 - - 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $- 1$ by $- 1$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 + 0 + 1 d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $d$ by $1$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 + 0 + d x \cdot 0}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $d x \cdot 0$ .

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Multiply $0$ by $d$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 + 0 + 0 x}{{(x^{2} - 4 x + 4)}^{2}}$

Multiply $0$ by $x$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 + 0 + 0}{{(x^{2} - 4 x + 4)}^{2}}$

Combine the opposite terms in $- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 + 0 + 0$ .

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Add $- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x$ and $0$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0 + 0}{{(x^{2} - 4 x + 4)}^{2}}$

Add $- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x$ and $0$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0 + 0}{{(x^{2} - 4 x + 4)}^{2}}$

Add $- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x$ and $0$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x + 0}{{(x^{2} - 4 x + 4)}^{2}}$

Add $- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x$ and $0$ .

$\frac{- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x}{{(x^{2} - 4 x + 4)}^{2}}$

Factor $x$ out of $- 2 x^{5} + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x$ .

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

$\frac{x (- 2 x^{4}) + 10 x^{4} - 17 x^{3} + 12 x^{2} - 4 x}{{(x^{2} - 4 x + 4)}^{2}}$

Factor $x$ out of $10 x^{4}$ .

$\frac{x (- 2 x^{4}) + x (10 x^{3}) - 17 x^{3} + 12 x^{2} - 4 x}{{(x^{2} - 4 x + 4)}^{2}}$

Factor $x$ out of $- 17 x^{3}$ .

$\frac{x (- 2 x^{4}) + x (10 x^{3}) + x (- 17 x^{2}) + 12 x^{2} - 4 x}{{(x^{2} - 4 x + 4)}^{2}}$

Factor $x$ out of $12 x^{2}$ .

$\frac{x (- 2 x^{4}) + x (10 x^{3}) + x (- 17 x^{2}) + x (12 x) - 4 x}{{(x^{2} - 4 x + 4)}^{2}}$

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

$\frac{x (- 2 x^{4}) + x (10 x^{3}) + x (- 17 x^{2}) + x (12 x) + x \cdot - 4}{{(x^{2} - 4 x + 4)}^{2}}$

Factor $x$ out of $x (- 2 x^{4}) + x (10 x^{3})$ .

$\frac{x (- 2 x^{4} + 10 x^{3}) + x (- 17 x^{2}) + x (12 x) + x \cdot - 4}{{(x^{2} - 4 x + 4)}^{2}}$

Factor $x$ out of $x (- 2 x^{4} + 10 x^{3}) + x (- 17 x^{2})$ .

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2}) + x (12 x) + x \cdot - 4}{{(x^{2} - 4 x + 4)}^{2}}$

Factor $x$ out of $x (- 2 x^{4} + 10 x^{3} - 17 x^{2}) + x (12 x)$ .

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x) + x \cdot - 4}{{(x^{2} - 4 x + 4)}^{2}}$

Factor $x$ out of $x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x) + x \cdot - 4$ .

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{{(x^{2} - 4 x + 4)}^{2}}$

Simplify the denominator.

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Factor using the perfect square rule.

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

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{{(x^{2} - 4 x + 2^{2})}^{2}}$

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

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

Rewrite the polynomial.

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{{(x^{2} - 2 \cdot x \cdot 2 + 2^{2})}^{2}}$

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

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{{({(x - 2)}^{2})}^{2}}$

Multiply the exponents in ${({(x - 2)}^{2})}^{2}$ .

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Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{{(x - 2)}^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{{(x - 2)}^{4}}$

Use the Binomial Theorem.

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{x^{4} + 4 x^{3} \cdot - 2 + 6 x^{2} {(- 2)}^{2} + 4 x {(- 2)}^{3} + {(- 2)}^{4}}$

Simplify each term.

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Multiply $- 2$ by $4$ .

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{x^{4} - 8 x^{3} + 6 x^{2} {(- 2)}^{2} + 4 x {(- 2)}^{3} + {(- 2)}^{4}}$

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

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{x^{4} - 8 x^{3} + 6 x^{2} \cdot 4 + 4 x {(- 2)}^{3} + {(- 2)}^{4}}$

Multiply $4$ by $6$ .

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{x^{4} - 8 x^{3} + 24 x^{2} + 4 x {(- 2)}^{3} + {(- 2)}^{4}}$

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

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{x^{4} - 8 x^{3} + 24 x^{2} + 4 x \cdot - 8 + {(- 2)}^{4}}$

Multiply $- 8$ by $4$ .

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{x^{4} - 8 x^{3} + 24 x^{2} - 32 x + {(- 2)}^{4}}$

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

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{x^{4} - 8 x^{3} + 24 x^{2} - 32 x + 16}$

Make each term match the terms from the binomial theorem formula.

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{x^{4} - 4 \cdot 2 x^{3} + 6 \cdot 2^{2} x^{2} - 4 \cdot 2^{3} x + 2^{4}}$

Factor $x^{4} - 4 \cdot 2 x^{3} + 6 \cdot 2^{2} x^{2} - 4 \cdot 2^{3} x + 2^{4}$ using the binomial theorem.

$\frac{x (- 2 x^{4} + 10 x^{3} - 17 x^{2} + 12 x - 4)}{{(2 - x)}^{4}}$

Factor $- 1$ out of $- 2 x^{4}$ .

$\frac{x (- (2 x^{4}) + 10 x^{3} - 17 x^{2} + 12 x - 4)}{{(2 - x)}^{4}}$

Factor $- 1$ out of $10 x^{3}$ .

$\frac{x (- (2 x^{4}) - (- 10 x^{3}) - 17 x^{2} + 12 x - 4)}{{(2 - x)}^{4}}$

Factor $- 1$ out of $- (2 x^{4}) - (- 10 x^{3})$ .

$\frac{x (- (2 x^{4} - 10 x^{3}) - 17 x^{2} + 12 x - 4)}{{(2 - x)}^{4}}$

Factor $- 1$ out of $- 17 x^{2}$ .

$\frac{x (- (2 x^{4} - 10 x^{3}) - (17 x^{2}) + 12 x - 4)}{{(2 - x)}^{4}}$

Factor $- 1$ out of $- (2 x^{4} - 10 x^{3}) - (17 x^{2})$ .

$\frac{x (- (2 x^{4} - 10 x^{3} + 17 x^{2}) + 12 x - 4)}{{(2 - x)}^{4}}$

Factor $- 1$ out of $12 x$ .

$\frac{x (- (2 x^{4} - 10 x^{3} + 17 x^{2}) - (- 12 x) - 4)}{{(2 - x)}^{4}}$

Factor $- 1$ out of $- (2 x^{4} - 10 x^{3} + 17 x^{2}) - (- 12 x)$ .

$\frac{x (- (2 x^{4} - 10 x^{3} + 17 x^{2} - 12 x) - 4)}{{(2 - x)}^{4}}$

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{x (- (2 x^{4} - 10 x^{3} + 17 x^{2} - 12 x) - 1 (4))}{{(2 - x)}^{4}}$

Factor $- 1$ out of $- (2 x^{4} - 10 x^{3} + 17 x^{2} - 12 x) - 1 (4)$ .

$\frac{x (- (2 x^{4} - 10 x^{3} + 17 x^{2} - 12 x + 4))}{{(2 - x)}^{4}}$

Rewrite $- (2 x^{4} - 10 x^{3} + 17 x^{2} - 12 x + 4)$ as $- 1 (2 x^{4} - 10 x^{3} + 17 x^{2} - 12 x + 4)$ .

$\frac{x (- 1 (2 x^{4} - 10 x^{3} + 17 x^{2} - 12 x + 4))}{{(2 - x)}^{4}}$

Move the negative in front of the fraction.

$- \frac{x (2 x^{4} - 10 x^{3} + 17 x^{2} - 12 x + 4)}{{(2 - x)}^{4}}$