Algebra Examples

$y = \frac{- 5 x^{3} - 5 x^{2} + 3}{- 5 x^{4} + 2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\frac{- 5 x^{3} - 5 x^{2} + 3}{- 5 x^{4} + 2})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Simplify terms.

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Step 3.1.1

Factor $- 1$ out of $- 5 x^{3}$ .

$\frac{d}{d y} [\frac{- (5 x^{3}) - 5 x^{2} + 3}{- 5 x^{4} + 2}]$

Step 3.1.2

Factor $- 1$ out of $- 5 x^{2}$ .

$\frac{d}{d y} [\frac{- (5 x^{3}) - (5 x^{2}) + 3}{- 5 x^{4} + 2}]$

Step 3.1.3

Factor $- 1$ out of $- (5 x^{3}) - (5 x^{2})$ .

$\frac{d}{d y} [\frac{- (5 x^{3} + 5 x^{2}) + 3}{- 5 x^{4} + 2}]$

Step 3.1.4

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{d}{d y} [\frac{- (5 x^{3} + 5 x^{2}) - 1 (- 3)}{- 5 x^{4} + 2}]$

Step 3.1.5

Factor $- 1$ out of $- (5 x^{3} + 5 x^{2}) - 1 (- 3)$ .

$\frac{d}{d y} [\frac{- (5 x^{3} + 5 x^{2} - 3)}{- 5 x^{4} + 2}]$

Step 3.1.6

Rewrite $- (5 x^{3} + 5 x^{2} - 3)$ as $- 1 (5 x^{3} + 5 x^{2} - 3)$ .

$\frac{d}{d y} [\frac{- 1 (5 x^{3} + 5 x^{2} - 3)}{- 5 x^{4} + 2}]$

Step 3.1.7

Factor $- 1$ out of $- 5 x^{4}$ .

$\frac{d}{d y} [\frac{- 1 (5 x^{3} + 5 x^{2} - 3)}{- (5 x^{4}) + 2}]$

Step 3.1.8

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{d}{d y} [\frac{- 1 (5 x^{3} + 5 x^{2} - 3)}{- (5 x^{4}) - 1 (- 2)}]$

Step 3.1.9

Factor $- 1$ out of $- (5 x^{4}) - 1 (- 2)$ .

$\frac{d}{d y} [\frac{- 1 (5 x^{3} + 5 x^{2} - 3)}{- (5 x^{4} - 2)}]$

Step 3.1.10

Rewrite $- (5 x^{4} - 2)$ as $- 1 (5 x^{4} - 2)$ .

$\frac{d}{d y} [\frac{- 1 (5 x^{3} + 5 x^{2} - 3)}{- 1 (5 x^{4} - 2)}]$

Step 3.1.11

Cancel the common factor.

$\frac{d}{d y} [\frac{- 1 (5 x^{3} + 5 x^{2} - 3)}{- 1 (5 x^{4} - 2)}]$

Step 3.1.12

Rewrite the expression.

$\frac{d}{d y} [\frac{5 x^{3} + 5 x^{2} - 3}{5 x^{4} - 2}]$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = 5 x^{3} + 5 x^{2} - 3$ and $g (y) = 5 x^{4} - 2$ .

$\frac{(5 x^{4} - 2) \frac{d}{d y} [5 x^{3} + 5 x^{2} - 3] - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $5 x^{3} + 5 x^{2} - 3$ with respect to $y$ is $\frac{d}{d y} [5 x^{3}] + \frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [- 3]$ .

$\frac{(5 x^{4} - 2) (\frac{d}{d y} [5 x^{3}] + \frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.3.2

Since $5$ is constant with respect to $y$ , the derivative of $5 x^{3}$ with respect to $y$ is $5 \frac{d}{d y} [x^{3}]$ .

$\frac{(5 x^{4} - 2) (5 \frac{d}{d y} [x^{3}] + \frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.4

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

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Step 3.4.1

To apply the Chain Rule, set $u_{1}$ as $x$ .

$\frac{(5 x^{4} - 2) (5 (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d y} [x]) + \frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.4.2

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

$\frac{(5 x^{4} - 2) (5 (3 {u_{1}}^{2} \frac{d}{d y} [x]) + \frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.4.3

Replace all occurrences of $u_{1}$ with $x$ .

$\frac{(5 x^{4} - 2) (5 (3 x^{2} \frac{d}{d y} [x]) + \frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.5

Multiply $3$ by $5$ .

$\frac{(5 x^{4} - 2) (15 (x^{2} \frac{d}{d y} [x]) + \frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.6

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + \frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.7

Since $5$ is constant with respect to $y$ , the derivative of $5 x^{2}$ with respect to $y$ is $5 \frac{d}{d y} [x^{2}]$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 5 \frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.8

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

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Step 3.8.1

To apply the Chain Rule, set $u_{2}$ as $x$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 5 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d y} [x]) + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.8.2

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

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 5 (2 u_{2} \frac{d}{d y} [x]) + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.8.3

Replace all occurrences of $u_{2}$ with $x$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 5 (2 x \frac{d}{d y} [x]) + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.9

Multiply $2$ by $5$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 (x \frac{d}{d y} [x]) + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.10

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x' + \frac{d}{d y} [- 3]) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.11

Differentiate.

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Step 3.11.1

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

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x' + 0) - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.11.2

Add $15 x^{2} x' + 10 x x'$ and $0$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - (5 x^{3} + 5 x^{2} - 3) \frac{d}{d y} [5 x^{4} - 2]}{{(5 x^{4} - 2)}^{2}}$

Step 3.11.3

By the Sum Rule, the derivative of $5 x^{4} - 2$ with respect to $y$ is $\frac{d}{d y} [5 x^{4}] + \frac{d}{d y} [- 2]$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - (5 x^{3} + 5 x^{2} - 3) (\frac{d}{d y} [5 x^{4}] + \frac{d}{d y} [- 2])}{{(5 x^{4} - 2)}^{2}}$

Step 3.11.4

Since $5$ is constant with respect to $y$ , the derivative of $5 x^{4}$ with respect to $y$ is $5 \frac{d}{d y} [x^{4}]$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - (5 x^{3} + 5 x^{2} - 3) (5 \frac{d}{d y} [x^{4}] + \frac{d}{d y} [- 2])}{{(5 x^{4} - 2)}^{2}}$

Step 3.12

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

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Step 3.12.1

To apply the Chain Rule, set $u_{3}$ as $x$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - (5 x^{3} + 5 x^{2} - 3) (5 (\frac{d}{d u_{3}} [{u_{3}}^{4}] \frac{d}{d y} [x]) + \frac{d}{d y} [- 2])}{{(5 x^{4} - 2)}^{2}}$

Step 3.12.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 4$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - (5 x^{3} + 5 x^{2} - 3) (5 (4 {u_{3}}^{3} \frac{d}{d y} [x]) + \frac{d}{d y} [- 2])}{{(5 x^{4} - 2)}^{2}}$

Step 3.12.3

Replace all occurrences of $u_{3}$ with $x$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - (5 x^{3} + 5 x^{2} - 3) (5 (4 x^{3} \frac{d}{d y} [x]) + \frac{d}{d y} [- 2])}{{(5 x^{4} - 2)}^{2}}$

Step 3.13

Multiply $4$ by $5$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - (5 x^{3} + 5 x^{2} - 3) (20 (x^{3} \frac{d}{d y} [x]) + \frac{d}{d y} [- 2])}{{(5 x^{4} - 2)}^{2}}$

Step 3.14

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - (5 x^{3} + 5 x^{2} - 3) (20 x^{3} x' + \frac{d}{d y} [- 2])}{{(5 x^{4} - 2)}^{2}}$

Step 3.15

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

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - (5 x^{3} + 5 x^{2} - 3) (20 x^{3} x' + 0)}{{(5 x^{4} - 2)}^{2}}$

Step 3.16

Simplify the expression.

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Step 3.16.1

Add $20 x^{3} x'$ and $0$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - (5 x^{3} + 5 x^{2} - 3) (20 x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.16.2

Multiply $20$ by $- 1$ .

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - 20 (5 x^{3} + 5 x^{2} - 3) (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17

Simplify.

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Step 3.17.1

Apply the distributive property.

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') + (- 20 (5 x^{3}) - 20 (5 x^{2}) - 20 \cdot - 3) (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.2

Apply the distributive property.

$\frac{(5 x^{4} - 2) (15 x^{2} x' + 10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3

Simplify the numerator.

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Step 3.17.3.1

Simplify each term.

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Step 3.17.3.1.1

Expand $(5 x^{4} - 2) (15 x^{2} x' + 10 x x')$ using the FOIL Method.

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Step 3.17.3.1.1.1

Apply the distributive property.

$\frac{5 x^{4} (15 x^{2} x' + 10 x x') - 2 (15 x^{2} x' + 10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.1.2

Apply the distributive property.

$\frac{5 x^{4} (15 x^{2} x') + 5 x^{4} (10 x x') - 2 (15 x^{2} x' + 10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.1.3

Apply the distributive property.

$\frac{5 x^{4} (15 x^{2} x') + 5 x^{4} (10 x x') - 2 (15 x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2

Simplify each term.

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Step 3.17.3.1.2.1

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

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Step 3.17.3.1.2.1.1

Move $x^{2}$ .

$\frac{5 (x^{2} x^{4}) (15 x') + 5 x^{4} (10 x x') - 2 (15 x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2.1.2

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

$\frac{5 x^{2 + 4} (15 x') + 5 x^{4} (10 x x') - 2 (15 x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2.1.3

Add $2$ and $4$ .

$\frac{5 x^{6} (15 x') + 5 x^{4} (10 x x') - 2 (15 x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2.2

Multiply $15$ by $5$ .

$\frac{75 x^{6} x' + 5 x^{4} (10 x x') - 2 (15 x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2.3

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

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Step 3.17.3.1.2.3.1

Move $x$ .

$\frac{75 x^{6} x' + 5 (x \cdot x^{4}) (10 x') - 2 (15 x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2.3.2

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

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Step 3.17.3.1.2.3.2.1

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

$\frac{75 x^{6} x' + 5 (x^{1} x^{4}) (10 x') - 2 (15 x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2.3.2.2

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

$\frac{75 x^{6} x' + 5 x^{1 + 4} (10 x') - 2 (15 x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2.3.3

Add $1$ and $4$ .

$\frac{75 x^{6} x' + 5 x^{5} (10 x') - 2 (15 x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2.4

Multiply $10$ by $5$ .

$\frac{75 x^{6} x' + 50 x^{5} x' - 2 (15 x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2.5

Multiply $15$ by $- 2$ .

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 (x^{2} x') - 2 (10 x x') - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.2.6

Multiply $10$ by $- 2$ .

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 20 (5 x^{3}) (x^{3} x') - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.3

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

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Step 3.17.3.1.3.1

Move $x^{3}$ .

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 20 (5 (x^{3} x^{3})) x' - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.3.2

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

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 20 (5 x^{3 + 3}) x' - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.3.3

Add $3$ and $3$ .

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 20 (5 x^{6}) x' - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.4

Multiply $5$ by $- 20$ .

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 100 x^{6} x' - 20 (5 x^{2}) (x^{3} x') - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.5

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

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Step 3.17.3.1.5.1

Move $x^{3}$ .

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 100 x^{6} x' - 20 (5 (x^{3} x^{2})) x' - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.5.2

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

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 100 x^{6} x' - 20 (5 x^{3 + 2}) x' - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.5.3

Add $3$ and $2$ .

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 100 x^{6} x' - 20 (5 x^{5}) x' - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.6

Multiply $5$ by $- 20$ .

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 100 x^{6} x' - 100 x^{5} x' - 20 \cdot - 3 (x^{3} x')}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.1.7

Multiply $- 20$ by $- 3$ .

$\frac{75 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 100 x^{6} x' - 100 x^{5} x' + 60 x^{3} x'}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.2

Subtract $100 x^{6} x'$ from $75 x^{6} x'$ .

$\frac{- 25 x^{6} x' + 50 x^{5} x' - 30 x^{2} x' - 20 x x' - 100 x^{5} x' + 60 x^{3} x'}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.3.3

Subtract $100 x^{5} x'$ from $50 x^{5} x'$ .

$\frac{- 25 x^{6} x' - 50 x^{5} x' - 30 x^{2} x' - 20 x x' + 60 x^{3} x'}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.4

Simplify the numerator.

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Step 3.17.4.1

Factor $5 x x'$ out of $- 25 x^{6} x' - 50 x^{5} x' - 30 x^{2} x' - 20 x x' + 60 x^{3} x'$ .

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Step 3.17.4.1.1

Factor $5 x x'$ out of $- 25 x^{6} x'$ .

$\frac{5 x x' (- 5 x^{5}) - 50 x^{5} x' - 30 x^{2} x' - 20 x x' + 60 x^{3} x'}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.4.1.2

Factor $5 x x'$ out of $- 50 x^{5} x'$ .

$\frac{5 x x' (- 5 x^{5}) + 5 x x' (- 10 x^{4}) - 30 x^{2} x' - 20 x x' + 60 x^{3} x'}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.4.1.3

Factor $5 x x'$ out of $- 30 x^{2} x'$ .

$\frac{5 x x' (- 5 x^{5}) + 5 x x' (- 10 x^{4}) + 5 x x' (- 6 x) - 20 x x' + 60 x^{3} x'}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.4.1.4

Factor $5 x x'$ out of $- 20 x x'$ .

$\frac{5 x x' (- 5 x^{5}) + 5 x x' (- 10 x^{4}) + 5 x x' (- 6 x) + 5 x x' (- 4) + 60 x^{3} x'}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.4.1.5

Factor $5 x x'$ out of $60 x^{3} x'$ .

$\frac{5 x x' (- 5 x^{5}) + 5 x x' (- 10 x^{4}) + 5 x x' (- 6 x) + 5 x x' (- 4) + 5 x x' (12 x^{2})}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.4.1.6

Factor $5 x x'$ out of $5 x x' (- 5 x^{5}) + 5 x x' (- 10 x^{4})$ .

$\frac{5 x x' (- 5 x^{5} - 10 x^{4}) + 5 x x' (- 6 x) + 5 x x' (- 4) + 5 x x' (12 x^{2})}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.4.1.7

Factor $5 x x'$ out of $5 x x' (- 5 x^{5} - 10 x^{4}) + 5 x x' (- 6 x)$ .

$\frac{5 x x' (- 5 x^{5} - 10 x^{4} - 6 x) + 5 x x' (- 4) + 5 x x' (12 x^{2})}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.4.1.8

Factor $5 x x'$ out of $5 x x' (- 5 x^{5} - 10 x^{4} - 6 x) + 5 x x' (- 4)$ .

$\frac{5 x x' (- 5 x^{5} - 10 x^{4} - 6 x - 4) + 5 x x' (12 x^{2})}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.4.1.9

Factor $5 x x'$ out of $5 x x' (- 5 x^{5} - 10 x^{4} - 6 x - 4) + 5 x x' (12 x^{2})$ .

$\frac{5 x x' (- 5 x^{5} - 10 x^{4} - 6 x - 4 + 12 x^{2})}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.4.2

Reorder terms.

$\frac{5 x x' (- 5 x^{5} - 10 x^{4} + 12 x^{2} - 6 x - 4)}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.5

Factor $- 1$ out of $- 5 x^{5}$ .

$\frac{5 x x' (- (5 x^{5}) - 10 x^{4} + 12 x^{2} - 6 x - 4)}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.6

Factor $- 1$ out of $- 10 x^{4}$ .

$\frac{5 x x' (- (5 x^{5}) - (10 x^{4}) + 12 x^{2} - 6 x - 4)}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.7

Factor $- 1$ out of $- (5 x^{5}) - (10 x^{4})$ .

$\frac{5 x x' (- (5 x^{5} + 10 x^{4}) + 12 x^{2} - 6 x - 4)}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.8

Factor $- 1$ out of $12 x^{2}$ .

$\frac{5 x x' (- (5 x^{5} + 10 x^{4}) - (- 12 x^{2}) - 6 x - 4)}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.9

Factor $- 1$ out of $- (5 x^{5} + 10 x^{4}) - (- 12 x^{2})$ .

$\frac{5 x x' (- (5 x^{5} + 10 x^{4} - 12 x^{2}) - 6 x - 4)}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.10

Factor $- 1$ out of $- 6 x$ .

$\frac{5 x x' (- (5 x^{5} + 10 x^{4} - 12 x^{2}) - (6 x) - 4)}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.11

Factor $- 1$ out of $- (5 x^{5} + 10 x^{4} - 12 x^{2}) - (6 x)$ .

$\frac{5 x x' (- (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x) - 4)}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.12

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

$\frac{5 x x' (- (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x) - 1 (4))}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.13

Factor $- 1$ out of $- (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x) - 1 (4)$ .

$\frac{5 x x' (- (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4))}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.14

Rewrite $- (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)$ as $- 1 (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)$ .

$\frac{5 x x' (- 1 (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4))}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.15

Move the negative in front of the fraction.

$- \frac{(5 x x') (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}{{(5 x^{4} - 2)}^{2}}$

Step 3.17.16

Reorder factors in $- \frac{(5 x x') (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}{{(5 x^{4} - 2)}^{2}}$ .

$- \frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$1 = - \frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}}$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $- \frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}} = 1$ .

$- \frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}} = 1$

Step 5.2

Divide each term in $- \frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}} = 1$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- \frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}} = 1$ by $- 1$ .

$\frac{- \frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}}}{- 1} = \frac{1}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}}}{1} = \frac{1}{- 1}$

Step 5.2.2.2

Simplify the expression.

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Step 5.2.2.2.1

Divide $\frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}}$ by $1$ .

$\frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}} = \frac{1}{- 1}$

Step 5.2.2.2.2

Reorder factors in $\frac{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) x'}{{(5 x^{4} - 2)}^{2}}$ .

$\frac{5 x x' (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}{{(5 x^{4} - 2)}^{2}} = \frac{1}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $1$ by $- 1$ .

$\frac{5 x x' (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}{{(5 x^{4} - 2)}^{2}} = - 1$

Step 5.3

Multiply both sides by ${(5 x^{4} - 2)}^{2}$ .

$\frac{5 x x' (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}{{(5 x^{4} - 2)}^{2}} {(5 x^{4} - 2)}^{2} = - {(5 x^{4} - 2)}^{2}$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $\frac{5 x x' (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}{{(5 x^{4} - 2)}^{2}} {(5 x^{4} - 2)}^{2}$ .

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Step 5.4.1.1

Cancel the common factor of ${(5 x^{4} - 2)}^{2}$ .

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Step 5.4.1.1.1

Cancel the common factor.

$\frac{5 x x' (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}{{(5 x^{4} - 2)}^{2}} {(5 x^{4} - 2)}^{2} = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.1.2

Rewrite the expression.

$5 x x' (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.2

Apply the distributive property.

$5 x x' (5 x^{5}) + 5 x x' (10 x^{4}) + 5 x x' (- 12 x^{2}) + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3

Simplify.

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Step 5.4.1.3.1

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 5.4.1.3.1.1

Move $x^{5}$ .

$5 (x^{5} x) x' \cdot 5 + 5 x x' (10 x^{4}) + 5 x x' (- 12 x^{2}) + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.1.2

Multiply $x^{5}$ by $x$ .

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Step 5.4.1.3.1.2.1

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

$5 (x^{5} x^{1}) x' \cdot 5 + 5 x x' (10 x^{4}) + 5 x x' (- 12 x^{2}) + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.1.2.2

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

$5 x^{5 + 1} x' \cdot 5 + 5 x x' (10 x^{4}) + 5 x x' (- 12 x^{2}) + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.1.3

Add $5$ and $1$ .

$5 x^{6} x' \cdot 5 + 5 x x' (10 x^{4}) + 5 x x' (- 12 x^{2}) + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.2

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

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Step 5.4.1.3.2.1

Move $x^{4}$ .

$5 x^{6} x' \cdot 5 + 5 (x^{4} x) x' \cdot 10 + 5 x x' (- 12 x^{2}) + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.2.2

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

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Step 5.4.1.3.2.2.1

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

$5 x^{6} x' \cdot 5 + 5 (x^{4} x^{1}) x' \cdot 10 + 5 x x' (- 12 x^{2}) + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.2.2.2

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

$5 x^{6} x' \cdot 5 + 5 x^{4 + 1} x' \cdot 10 + 5 x x' (- 12 x^{2}) + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.2.3

Add $4$ and $1$ .

$5 x^{6} x' \cdot 5 + 5 x^{5} x' \cdot 10 + 5 x x' (- 12 x^{2}) + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.3

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

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Step 5.4.1.3.3.1

Move $x^{2}$ .

$5 x^{6} x' \cdot 5 + 5 x^{5} x' \cdot 10 + 5 (x^{2} x) x' \cdot - 12 + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.3.2

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

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Step 5.4.1.3.3.2.1

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

$5 x^{6} x' \cdot 5 + 5 x^{5} x' \cdot 10 + 5 (x^{2} x^{1}) x' \cdot - 12 + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.3.2.2

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

$5 x^{6} x' \cdot 5 + 5 x^{5} x' \cdot 10 + 5 x^{2 + 1} x' \cdot - 12 + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.3.3

Add $2$ and $1$ .

$5 x^{6} x' \cdot 5 + 5 x^{5} x' \cdot 10 + 5 x^{3} x' \cdot - 12 + 5 x x' (6 x) + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.4

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

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Step 5.4.1.3.4.1

Move $x$ .

$5 x^{6} x' \cdot 5 + 5 x^{5} x' \cdot 10 + 5 x^{3} x' \cdot - 12 + 5 (x \cdot x) x' \cdot 6 + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.4.2

Multiply $x$ by $x$ .

$5 x^{6} x' \cdot 5 + 5 x^{5} x' \cdot 10 + 5 x^{3} x' \cdot - 12 + 5 x^{2} x' \cdot 6 + 5 x x' \cdot 4 = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.3.5

Multiply $4$ by $5$ .

$5 x^{6} x' \cdot 5 + 5 x^{5} x' \cdot 10 + 5 x^{3} x' \cdot - 12 + 5 x^{2} x' \cdot 6 + 20 x x' = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.4

Simplify each term.

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Step 5.4.1.4.1

Multiply $5$ by $5$ .

$25 x^{6} x' + 5 x^{5} x' \cdot 10 + 5 x^{3} x' \cdot - 12 + 5 x^{2} x' \cdot 6 + 20 x x' = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.4.2

Multiply $10$ by $5$ .

$25 x^{6} x' + 50 x^{5} x' + 5 x^{3} x' \cdot - 12 + 5 x^{2} x' \cdot 6 + 20 x x' = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.4.3

Multiply $- 12$ by $5$ .

$25 x^{6} x' + 50 x^{5} x' - 60 x^{3} x' + 5 x^{2} x' \cdot 6 + 20 x x' = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.4.4

Multiply $6$ by $5$ .

$25 x^{6} x' + 50 x^{5} x' - 60 x^{3} x' + 30 x^{2} x' + 20 x x' = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.5

Reorder.

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Step 5.4.1.5.1

Move $x^{6}$ .

$25 x' x^{6} + 50 x^{5} x' - 60 x^{3} x' + 30 x^{2} x' + 20 x x' = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.5.2

Move $x^{5}$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x^{3} x' + 30 x^{2} x' + 20 x x' = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.5.3

Move $x^{3}$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x^{2} x' + 20 x x' = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.5.4

Move $x^{2}$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x x' = - {(5 x^{4} - 2)}^{2}$

Step 5.4.1.5.5

Move $x$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - {(5 x^{4} - 2)}^{2}$

Step 5.5

Solve for $x'$ .

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Step 5.5.1

Simplify $- {(5 x^{4} - 2)}^{2}$ .

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Step 5.5.1.1

Rewrite ${(5 x^{4} - 2)}^{2}$ as $(5 x^{4} - 2) (5 x^{4} - 2)$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - ((5 x^{4} - 2) (5 x^{4} - 2))$

Step 5.5.1.2

Expand $(5 x^{4} - 2) (5 x^{4} - 2)$ using the FOIL Method.

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Step 5.5.1.2.1

Apply the distributive property.

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (5 x^{4} (5 x^{4} - 2) - 2 (5 x^{4} - 2))$

Step 5.5.1.2.2

Apply the distributive property.

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (5 x^{4} (5 x^{4}) + 5 x^{4} \cdot - 2 - 2 (5 x^{4} - 2))$

Step 5.5.1.2.3

Apply the distributive property.

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (5 x^{4} (5 x^{4}) + 5 x^{4} \cdot - 2 - 2 (5 x^{4}) - 2 \cdot - 2)$

Step 5.5.1.3

Simplify and combine like terms.

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Step 5.5.1.3.1

Simplify each term.

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Step 5.5.1.3.1.1

Rewrite using the commutative property of multiplication.

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (5 \cdot 5 x^{4} x^{4} + 5 x^{4} \cdot - 2 - 2 (5 x^{4}) - 2 \cdot - 2)$

Step 5.5.1.3.1.2

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

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Step 5.5.1.3.1.2.1

Move $x^{4}$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (5 \cdot 5 (x^{4} x^{4}) + 5 x^{4} \cdot - 2 - 2 (5 x^{4}) - 2 \cdot - 2)$

Step 5.5.1.3.1.2.2

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

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (5 \cdot 5 x^{4 + 4} + 5 x^{4} \cdot - 2 - 2 (5 x^{4}) - 2 \cdot - 2)$

Step 5.5.1.3.1.2.3

Add $4$ and $4$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (5 \cdot 5 x^{8} + 5 x^{4} \cdot - 2 - 2 (5 x^{4}) - 2 \cdot - 2)$

Step 5.5.1.3.1.3

Multiply $5$ by $5$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (25 x^{8} + 5 x^{4} \cdot - 2 - 2 (5 x^{4}) - 2 \cdot - 2)$

Step 5.5.1.3.1.4

Multiply $- 2$ by $5$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (25 x^{8} - 10 x^{4} - 2 (5 x^{4}) - 2 \cdot - 2)$

Step 5.5.1.3.1.5

Multiply $5$ by $- 2$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (25 x^{8} - 10 x^{4} - 10 x^{4} - 2 \cdot - 2)$

Step 5.5.1.3.1.6

Multiply $- 2$ by $- 2$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (25 x^{8} - 10 x^{4} - 10 x^{4} + 4)$

Step 5.5.1.3.2

Subtract $10 x^{4}$ from $- 10 x^{4}$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (25 x^{8} - 20 x^{4} + 4)$

Step 5.5.1.4

Apply the distributive property.

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - (25 x^{8}) - (- 20 x^{4}) - 1 \cdot 4$

Step 5.5.1.5

Simplify.

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Step 5.5.1.5.1

Multiply $25$ by $- 1$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - 25 x^{8} - (- 20 x^{4}) - 1 \cdot 4$

Step 5.5.1.5.2

Multiply $- 20$ by $- 1$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - 25 x^{8} + 20 x^{4} - 1 \cdot 4$

Step 5.5.1.5.3

Multiply $- 1$ by $4$ .

$25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - 25 x^{8} + 20 x^{4} - 4$

Step 5.5.2

Factor $5 x' x$ out of $25 x' x^{6} + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x$ .

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Step 5.5.2.1

Factor $5 x' x$ out of $25 x' x^{6}$ .

$5 x' x (5 x^{5}) + 50 x' x^{5} - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - 25 x^{8} + 20 x^{4} - 4$

Step 5.5.2.2

Factor $5 x' x$ out of $50 x' x^{5}$ .

$5 x' x (5 x^{5}) + 5 x' x (10 x^{4}) - 60 x' x^{3} + 30 x' x^{2} + 20 x' x = - 25 x^{8} + 20 x^{4} - 4$

Step 5.5.2.3

Factor $5 x' x$ out of $- 60 x' x^{3}$ .

$5 x' x (5 x^{5}) + 5 x' x (10 x^{4}) + 5 x' x (- 12 x^{2}) + 30 x' x^{2} + 20 x' x = - 25 x^{8} + 20 x^{4} - 4$

Step 5.5.2.4

Factor $5 x' x$ out of $30 x' x^{2}$ .

$5 x' x (5 x^{5}) + 5 x' x (10 x^{4}) + 5 x' x (- 12 x^{2}) + 5 x' x (6 x) + 20 x' x = - 25 x^{8} + 20 x^{4} - 4$

Step 5.5.2.5

Factor $5 x' x$ out of $20 x' x$ .

$5 x' x (5 x^{5}) + 5 x' x (10 x^{4}) + 5 x' x (- 12 x^{2}) + 5 x' x (6 x) + 5 x' x \cdot 4 = - 25 x^{8} + 20 x^{4} - 4$

Step 5.5.2.6

Factor $5 x' x$ out of $5 x' x (5 x^{5}) + 5 x' x (10 x^{4})$ .

$5 x' x (5 x^{5} + 10 x^{4}) + 5 x' x (- 12 x^{2}) + 5 x' x (6 x) + 5 x' x \cdot 4 = - 25 x^{8} + 20 x^{4} - 4$

Step 5.5.2.7

Factor $5 x' x$ out of $5 x' x (5 x^{5} + 10 x^{4}) + 5 x' x (- 12 x^{2})$ .

$5 x' x (5 x^{5} + 10 x^{4} - 12 x^{2}) + 5 x' x (6 x) + 5 x' x \cdot 4 = - 25 x^{8} + 20 x^{4} - 4$

Step 5.5.2.8

Factor $5 x' x$ out of $5 x' x (5 x^{5} + 10 x^{4} - 12 x^{2}) + 5 x' x (6 x)$ .

$5 x' x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x) + 5 x' x \cdot 4 = - 25 x^{8} + 20 x^{4} - 4$

Step 5.5.2.9

Factor $5 x' x$ out of $5 x' x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x) + 5 x' x \cdot 4$ .

$5 x' x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) = - 25 x^{8} + 20 x^{4} - 4$

Step 5.5.3

Divide each term in $5 x' x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) = - 25 x^{8} + 20 x^{4} - 4$ by $5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)$ and simplify.

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Step 5.5.3.1

Divide each term in $5 x' x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) = - 25 x^{8} + 20 x^{4} - 4$ by $5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)$ .

$\frac{5 x' x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} = \frac{- 25 x^{8}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.2

Simplify the left side.

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Step 5.5.3.2.1

Cancel the common factor of $5$ .

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Step 5.5.3.2.1.1

Cancel the common factor.

Step 5.5.3.2.1.2

Rewrite the expression.

$\frac{x' x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}{x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} = \frac{- 25 x^{8}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.2.2

Cancel the common factor of $x$ .

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Step 5.5.3.2.2.1

Cancel the common factor.

Step 5.5.3.2.2.2

Rewrite the expression.

$\frac{x' (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} = \frac{- 25 x^{8}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.2.3

Cancel the common factor of $5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4$ .

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Step 5.5.3.2.3.1

Cancel the common factor.

Step 5.5.3.2.3.2

Divide $x'$ by $1$ .

$x' = \frac{- 25 x^{8}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3

Simplify the right side.

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Step 5.5.3.3.1

Simplify each term.

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Step 5.5.3.3.1.1

Cancel the common factor of $- 25$ and $5$ .

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Step 5.5.3.3.1.1.1

Factor $5$ out of $- 25 x^{8}$ .

$x' = \frac{5 (- 5 x^{8})}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.1.2

Cancel the common factors.

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Step 5.5.3.3.1.1.2.1

Factor $5$ out of $5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)$ .

$x' = \frac{5 (- 5 x^{8})}{5 (x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4))} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.1.2.2

Cancel the common factor.

Step 5.5.3.3.1.1.2.3

Rewrite the expression.

$x' = \frac{- 5 x^{8}}{x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.2

Cancel the common factor of $x^{8}$ and $x$ .

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Step 5.5.3.3.1.2.1

Factor $x$ out of $- 5 x^{8}$ .

$x' = \frac{x (- 5 x^{7})}{x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.2.2

Cancel the common factors.

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Step 5.5.3.3.1.2.2.1

Cancel the common factor.

$x' = \frac{x (- 5 x^{7})}{x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.2.2.2

Rewrite the expression.

$x' = \frac{- 5 x^{7}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.3

Move the negative in front of the fraction.

$x' = - \frac{5 x^{7}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{20 x^{4}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.4

Cancel the common factor of $20$ and $5$ .

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Step 5.5.3.3.1.4.1

Factor $5$ out of $20 x^{4}$ .

$x' = - \frac{5 x^{7}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{5 (4 x^{4})}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.4.2

Cancel the common factors.

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Step 5.5.3.3.1.4.2.1

Factor $5$ out of $5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)$ .

$x' = - \frac{5 x^{7}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{5 (4 x^{4})}{5 (x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4))} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.4.2.2

Cancel the common factor.

$x' = - \frac{5 x^{7}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{5 (4 x^{4})}{5 (x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4))} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.4.2.3

Rewrite the expression.

$x' = - \frac{5 x^{7}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{4 x^{4}}{x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.5

Cancel the common factor of $x^{4}$ and $x$ .

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Step 5.5.3.3.1.5.1

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

$x' = - \frac{5 x^{7}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{x (4 x^{3})}{x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.5.2

Cancel the common factors.

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Step 5.5.3.3.1.5.2.1

Cancel the common factor.

$x' = - \frac{5 x^{7}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{x (4 x^{3})}{x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.5.2.2

Rewrite the expression.

$x' = - \frac{5 x^{7}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{4 x^{3}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{- 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.1.6

Move the negative in front of the fraction.

$x' = - \frac{5 x^{7}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} + \frac{4 x^{3}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} - \frac{4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.2

Simplify terms.

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Step 5.5.3.3.2.1

Combine the numerators over the common denominator.

$x' = \frac{- 5 x^{7} + 4 x^{3}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} - \frac{4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.2.2

Factor $x^{3}$ out of $- 5 x^{7} + 4 x^{3}$ .

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Step 5.5.3.3.2.2.1

Factor $x^{3}$ out of $- 5 x^{7}$ .

$x' = \frac{x^{3} (- 5 x^{4}) + 4 x^{3}}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} - \frac{4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.2.2.2

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

$x' = \frac{x^{3} (- 5 x^{4}) + x^{3} \cdot 4}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} - \frac{4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.2.2.3

Factor $x^{3}$ out of $x^{3} (- 5 x^{4}) + x^{3} \cdot 4$ .

$x' = \frac{x^{3} (- 5 x^{4} + 4)}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} - \frac{4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.3

To write $\frac{x^{3} (- 5 x^{4} + 4)}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4}$ as a fraction with a common denominator, multiply by $\frac{5 x}{5 x}$ .

$x' = \frac{x^{3} (- 5 x^{4} + 4)}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4} \cdot \frac{5 x}{5 x} - \frac{4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.4

Write each expression with a common denominator of $(5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) \cdot 5 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.3.4.1

Multiply $\frac{x^{3} (- 5 x^{4} + 4)}{5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4}$ by $\frac{5 x}{5 x}$ .

$x' = \frac{x^{3} (- 5 x^{4} + 4) (5 x)}{(5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) (5 x)} - \frac{4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.4.2

Reorder the factors of $(5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4) (5 x)$ .

$x' = \frac{x^{3} (- 5 x^{4} + 4) (5 x)}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)} - \frac{4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.5

Combine the numerators over the common denominator.

$x' = \frac{x^{3} (- 5 x^{4} + 4) (5 x) - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6

Simplify the numerator.

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Step 5.5.3.3.6.1

Rewrite using the commutative property of multiplication.

$x' = \frac{5 x^{3} (- 5 x^{4} + 4) x - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.2

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

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Step 5.5.3.3.6.2.1

Move $x$ .

$x' = \frac{5 (x \cdot x^{3}) (- 5 x^{4} + 4) - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.2.2

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

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Step 5.5.3.3.6.2.2.1

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

$x' = \frac{5 (x^{1} x^{3}) (- 5 x^{4} + 4) - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.2.2.2

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

$x' = \frac{5 x^{1 + 3} (- 5 x^{4} + 4) - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.2.3

Add $1$ and $3$ .

$x' = \frac{5 x^{4} (- 5 x^{4} + 4) - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.3

Apply the distributive property.

$x' = \frac{5 x^{4} (- 5 x^{4}) + 5 x^{4} \cdot 4 - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.4

Rewrite using the commutative property of multiplication.

$x' = \frac{5 \cdot - 5 x^{4} x^{4} + 5 x^{4} \cdot 4 - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.5

Multiply $4$ by $5$ .

$x' = \frac{5 \cdot - 5 x^{4} x^{4} + 20 x^{4} - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.6

Simplify each term.

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Step 5.5.3.3.6.6.1

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

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Step 5.5.3.3.6.6.1.1

Move $x^{4}$ .

$x' = \frac{5 \cdot - 5 (x^{4} x^{4}) + 20 x^{4} - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.6.1.2

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

$x' = \frac{5 \cdot - 5 x^{4 + 4} + 20 x^{4} - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.6.1.3

Add $4$ and $4$ .

$x' = \frac{5 \cdot - 5 x^{8} + 20 x^{4} - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.6.2

Multiply $5$ by $- 5$ .

$x' = \frac{- 25 x^{8} + 20 x^{4} - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.7

Rewrite $- 25 x^{8} + 20 x^{4} - 4$ in a factored form.

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Step 5.5.3.3.6.7.1

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$x' = \frac{- 25 {(x^{4})}^{2} + 20 x^{4} - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.7.2

Let $u = x^{4}$ . Substitute $u$ for all occurrences of $x^{4}$ .

$x' = \frac{- 25 u^{2} + 20 u - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.7.3

Factor by grouping.

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Step 5.5.3.3.6.7.3.1

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 $a \cdot c = - 25 \cdot - 4 = 100$ and whose sum is $b = 20$ .

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Step 5.5.3.3.6.7.3.1.1

Factor $20$ out of $20 u$ .

$x' = \frac{- 25 u^{2} + 20 (u) - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.7.3.1.2

Rewrite $20$ as $10$ plus $10$

$x' = \frac{- 25 u^{2} + (10 + 10) u - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.7.3.1.3

Apply the distributive property.

$x' = \frac{- 25 u^{2} + 10 u + 10 u - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.7.3.2

Factor out the greatest common factor from each group.

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Step 5.5.3.3.6.7.3.2.1

Group the first two terms and the last two terms.

$x' = \frac{(- 25 u^{2} + 10 u) + 10 u - 4}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.7.3.2.2

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

$x' = \frac{5 u (- 5 u + 2) - 2 (- 5 u + 2)}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.7.3.3

Factor the polynomial by factoring out the greatest common factor, $- 5 u + 2$ .

$x' = \frac{(- 5 u + 2) (5 u - 2)}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.6.7.4

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

$x' = \frac{(- 5 x^{4} + 2) (5 x^{4} - 2)}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.7

Simplify the numerator.

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Step 5.5.3.3.7.1

Factor $- 1$ out of $- 5 x^{4}$ .

$x' = \frac{(- (5 x^{4}) + 2) (5 x^{4} - 2)}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.7.2

Rewrite $2$ as $- 1 (- 2)$ .

$x' = \frac{(- (5 x^{4}) - 1 (- 2)) (5 x^{4} - 2)}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.7.3

Factor $- 1$ out of $- (5 x^{4}) - 1 (- 2)$ .

$x' = \frac{- (5 x^{4} - 2) (5 x^{4} - 2)}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.7.4

Rewrite $- (5 x^{4} - 2)$ as $- 1 (5 x^{4} - 2)$ .

$x' = \frac{- 1 (5 x^{4} - 2) (5 x^{4} - 2)}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.7.5

Raise $5 x^{4} - 2$ to the power of $1$ .

$x' = \frac{- 1 ({(5 x^{4} - 2)}^{1} (5 x^{4} - 2))}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.7.6

Raise $5 x^{4} - 2$ to the power of $1$ .

$x' = \frac{- 1 ({(5 x^{4} - 2)}^{1} {(5 x^{4} - 2)}^{1})}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.7.7

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

$x' = \frac{- 1 {(5 x^{4} - 2)}^{1 + 1}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.7.8

Add $1$ and $1$ .

$x' = \frac{- 1 {(5 x^{4} - 2)}^{2}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 5.5.3.3.8

Move the negative in front of the fraction.

$x' = - \frac{{(5 x^{4} - 2)}^{2}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$

Step 6

Replace $x'$ with $\frac{d x}{d y}$ .

$\frac{d x}{d y} = - \frac{{(5 x^{4} - 2)}^{2}}{5 x (5 x^{5} + 10 x^{4} - 12 x^{2} + 6 x + 4)}$