Finite Math Examples

$\frac{\frac{x + 4}{x - 6} - \frac{x - 1}{x + 5}}{\frac{x}{x + 1} - \frac{3 x - 1}{x}}$

Step 1

Multiply the numerator and denominator of the fraction by $(x - 6) (x + 5) (x + 1) x$ .

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Step 1.1

Multiply $\frac{\frac{x + 4}{x - 6} - \frac{x - 1}{x + 5}}{\frac{x}{x + 1} - \frac{3 x - 1}{x}}$ by $\frac{(x - 6) (x + 5) (x + 1) x}{(x - 6) (x + 5) (x + 1) x}$ .

$\frac{(x - 6) (x + 5) (x + 1) x}{(x - 6) (x + 5) (x + 1) x} \cdot \frac{\frac{x + 4}{x - 6} - \frac{x - 1}{x + 5}}{\frac{x}{x + 1} - \frac{3 x - 1}{x}}$

Step 1.2

Combine.

$\frac{(x - 6) (x + 5) (x + 1) x (\frac{x + 4}{x - 6} - \frac{x - 1}{x + 5})}{(x - 6) (x + 5) (x + 1) x (\frac{x}{x + 1} - \frac{3 x - 1}{x})}$

Step 2

Apply the distributive property.

$\frac{(x - 6) (x + 5) (x + 1) x \frac{x + 4}{x - 6} + (x - 6) (x + 5) (x + 1) x (- \frac{x - 1}{x + 5})}{(x - 6) (x + 5) (x + 1) x \frac{x}{x + 1} + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x - 6$ .

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Step 3.1.1

Factor $x - 6$ out of $(x - 6) (x + 5) (x + 1) x$ .

$\frac{(x - 6) (((x + 5) (x + 1)) x) \frac{x + 4}{x - 6} + (x - 6) (x + 5) (x + 1) x (- \frac{x - 1}{x + 5})}{(x - 6) (x + 5) (x + 1) x \frac{x}{x + 1} + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.1.2

Cancel the common factor.

Step 3.1.3

Rewrite the expression.

$\frac{((x + 5) (x + 1)) x (x + 4) + (x - 6) (x + 5) (x + 1) x (- \frac{x - 1}{x + 5})}{(x - 6) (x + 5) (x + 1) x \frac{x}{x + 1} + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.2

Cancel the common factor of $x + 5$ .

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Step 3.2.1

Move the leading negative in $- \frac{x - 1}{x + 5}$ into the numerator.

$\frac{((x + 5) (x + 1)) x (x + 4) + (x - 6) (x + 5) (x + 1) x \frac{- (x - 1)}{x + 5}}{(x - 6) (x + 5) (x + 1) x \frac{x}{x + 1} + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.2.2

Factor $x + 5$ out of $(x - 6) (x + 5) (x + 1) x$ .

$\frac{((x + 5) (x + 1)) x (x + 4) + (x + 5) (((x - 6) (x + 1)) x) \frac{- (x - 1)}{x + 5}}{(x - 6) (x + 5) (x + 1) x \frac{x}{x + 1} + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.2.3

Cancel the common factor.

$\frac{((x + 5) (x + 1)) x (x + 4) + (x + 5) (((x - 6) (x + 1)) x) \frac{- (x - 1)}{x + 5}}{(x - 6) (x + 5) (x + 1) x \frac{x}{x + 1} + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.2.4

Rewrite the expression.

$\frac{((x + 5) (x + 1)) x (x + 4) + ((x - 6) (x + 1)) x (- (x - 1))}{(x - 6) (x + 5) (x + 1) x \frac{x}{x + 1} + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.3

Cancel the common factor of $x + 1$ .

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Step 3.3.1

Factor $x + 1$ out of $(x - 6) (x + 5) (x + 1) x$ .

$\frac{((x + 5) (x + 1)) x (x + 4) + ((x - 6) (x + 1)) x (- (x - 1))}{(x + 1) ((x - 6) (x + 5) x) \frac{x}{x + 1} + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.3.2

Cancel the common factor.

$\frac{((x + 5) (x + 1)) x (x + 4) + ((x - 6) (x + 1)) x (- (x - 1))}{(x + 1) ((x - 6) (x + 5) x) \frac{x}{x + 1} + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.3.3

Rewrite the expression.

$\frac{((x + 5) (x + 1)) x (x + 4) + ((x - 6) (x + 1)) x (- (x - 1))}{(x - 6) (x + 5) x \cdot x + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.4

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

$\frac{((x + 5) (x + 1)) x (x + 4) + ((x - 6) (x + 1)) x (- (x - 1))}{(x - 6) (x + 5) (x^{1} x) + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.5

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

$\frac{((x + 5) (x + 1)) x (x + 4) + ((x - 6) (x + 1)) x (- (x - 1))}{(x - 6) (x + 5) (x^{1} x^{1}) + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.6

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

$\frac{((x + 5) (x + 1)) x (x + 4) + ((x - 6) (x + 1)) x (- (x - 1))}{(x - 6) (x + 5) x^{1 + 1} + (x - 6) (x + 5) (x + 1) x (- \frac{3 x - 1}{x})}$

Step 3.7

Add $1$ and $1$ .

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

Step 3.8

Cancel the common factor of $x$ .

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Step 3.8.1

Move the leading negative in $- \frac{3 x - 1}{x}$ into the numerator.

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

Step 3.8.2

Factor $x$ out of $(x - 6) (x + 5) (x + 1) x$ .

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

Step 3.8.3

Cancel the common factor.

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

Step 3.8.4

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $(x + 1) x$ out of $((x + 5) (x + 1)) x (x + 4) + ((x - 6) (x + 1)) x (- (x - 1))$ .

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Step 4.1.1

Factor $(x + 1) x$ out of $((x + 5) (x + 1)) x (x + 4)$ .

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

Step 4.1.2

Factor $(x + 1) x$ out of $((x - 6) (x + 1)) x (- (x - 1))$ .

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

Step 4.1.3

Factor $(x + 1) x$ out of $(x + 1) x ((x + 5) (x + 4)) + (x + 1) x ((x - 6) (- (x - 1)))$ .

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

Step 4.2

Expand $(x + 5) (x + 4)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

$\frac{(x + 1) x (x \cdot x + x \cdot 4 + 5 (x + 4) + (x - 6) (- (x - 1)))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.2.3

Apply the distributive property.

$\frac{(x + 1) x (x \cdot x + x \cdot 4 + 5 x + 5 \cdot 4 + (x - 6) (- (x - 1)))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $x$ by $x$ .

$\frac{(x + 1) x (x^{2} + x \cdot 4 + 5 x + 5 \cdot 4 + (x - 6) (- (x - 1)))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.3.1.2

Move $4$ to the left of $x$ .

$\frac{(x + 1) x (x^{2} + 4 \cdot x + 5 x + 5 \cdot 4 + (x - 6) (- (x - 1)))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.3.1.3

Multiply $5$ by $4$ .

$\frac{(x + 1) x (x^{2} + 4 x + 5 x + 20 + (x - 6) (- (x - 1)))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.3.2

Add $4 x$ and $5 x$ .

$\frac{(x + 1) x (x^{2} + 9 x + 20 + (x - 6) (- (x - 1)))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.4

Apply the distributive property.

$\frac{(x + 1) x (x^{2} + 9 x + 20 + (x - 6) (- x - - 1))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.5

Multiply $- 1$ by $- 1$ .

$\frac{(x + 1) x (x^{2} + 9 x + 20 + (x - 6) (- x + 1))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.6

Expand $(x - 6) (- x + 1)$ using the FOIL Method.

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Step 4.6.1

Apply the distributive property.

$\frac{(x + 1) x (x^{2} + 9 x + 20 + x (- x + 1) - 6 (- x + 1))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.6.2

Apply the distributive property.

$\frac{(x + 1) x (x^{2} + 9 x + 20 + x (- x) + x \cdot 1 - 6 (- x + 1))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.6.3

Apply the distributive property.

$\frac{(x + 1) x (x^{2} + 9 x + 20 + x (- x) + x \cdot 1 - 6 (- x) - 6 \cdot 1)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.7

Simplify and combine like terms.

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Step 4.7.1

Simplify each term.

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Step 4.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{(x + 1) x (x^{2} + 9 x + 20 - x \cdot x + x \cdot 1 - 6 (- x) - 6 \cdot 1)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.7.1.2

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

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Step 4.7.1.2.1

Move $x$ .

$\frac{(x + 1) x (x^{2} + 9 x + 20 - (x \cdot x) + x \cdot 1 - 6 (- x) - 6 \cdot 1)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.7.1.2.2

Multiply $x$ by $x$ .

$\frac{(x + 1) x (x^{2} + 9 x + 20 - x^{2} + x \cdot 1 - 6 (- x) - 6 \cdot 1)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.7.1.3

Multiply $x$ by $1$ .

$\frac{(x + 1) x (x^{2} + 9 x + 20 - x^{2} + x - 6 (- x) - 6 \cdot 1)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.7.1.4

Multiply $- 1$ by $- 6$ .

$\frac{(x + 1) x (x^{2} + 9 x + 20 - x^{2} + x + 6 x - 6 \cdot 1)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.7.1.5

Multiply $- 6$ by $1$ .

$\frac{(x + 1) x (x^{2} + 9 x + 20 - x^{2} + x + 6 x - 6)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.7.2

Add $x$ and $6 x$ .

$\frac{(x + 1) x (x^{2} + 9 x + 20 - x^{2} + 7 x - 6)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.8

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

$\frac{(x + 1) x (9 x + 20 + 0 + 7 x - 6)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.9

Add $9 x$ and $0$ .

$\frac{(x + 1) x (9 x + 20 + 7 x - 6)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.10

Add $9 x$ and $7 x$ .

$\frac{(x + 1) x (16 x + 20 - 6)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.11

Subtract $6$ from $20$ .

$\frac{(x + 1) x (16 x + 14)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.12

Factor $2$ out of $16 x + 14$ .

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Step 4.12.1

Factor $2$ out of $16 x$ .

$\frac{(x + 1) x (2 (8 x) + 14)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.12.2

Factor $2$ out of $14$ .

$\frac{(x + 1) x (2 (8 x) + 2 (7))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 4.12.3

Factor $2$ out of $2 (8 x) + 2 (7)$ .

$\frac{(x + 1) x (2 (8 x + 7))}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $(x - 6) (x + 5)$ out of $(x - 6) (x + 5) x^{2} + (x - 6) (x + 5) (x + 1) (- (3 x - 1))$ .

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Step 5.1.1

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

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2}) + (x - 6) (x + 5) (x + 1) (- (3 x - 1))}$

Step 5.1.2

Factor $(x - 6) (x + 5)$ out of $(x - 6) (x + 5) (x + 1) (- (3 x - 1))$ .

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2}) + (x - 6) (x + 5) ((x + 1) (- (3 x - 1)))}$

Step 5.1.3

Factor $(x - 6) (x + 5)$ out of $(x - 6) (x + 5) (x^{2}) + (x - 6) (x + 5) ((x + 1) (- (3 x - 1)))$ .

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} + (x + 1) (- (3 x - 1)))}$

Step 5.2

Apply the distributive property.

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} + (x + 1) (- (3 x) - - 1))}$

Step 5.3

Multiply $3$ by $- 1$ .

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} + (x + 1) (- 3 x - - 1))}$

Step 5.4

Multiply $- 1$ by $- 1$ .

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} + (x + 1) (- 3 x + 1))}$

Step 5.5

Expand $(x + 1) (- 3 x + 1)$ using the FOIL Method.

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Step 5.5.1

Apply the distributive property.

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} + x (- 3 x + 1) + 1 (- 3 x + 1))}$

Step 5.5.2

Apply the distributive property.

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} + x (- 3 x) + x \cdot 1 + 1 (- 3 x + 1))}$

Step 5.5.3

Apply the distributive property.

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} + x (- 3 x) + x \cdot 1 + 1 (- 3 x) + 1 \cdot 1)}$

Step 5.6

Simplify and combine like terms.

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Step 5.6.1

Simplify each term.

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Step 5.6.1.1

Rewrite using the commutative property of multiplication.

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} - 3 x \cdot x + x \cdot 1 + 1 (- 3 x) + 1 \cdot 1)}$

Step 5.6.1.2

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

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Step 5.6.1.2.1

Move $x$ .

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} - 3 (x \cdot x) + x \cdot 1 + 1 (- 3 x) + 1 \cdot 1)}$

Step 5.6.1.2.2

Multiply $x$ by $x$ .

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} - 3 x^{2} + x \cdot 1 + 1 (- 3 x) + 1 \cdot 1)}$

Step 5.6.1.3

Multiply $x$ by $1$ .

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} - 3 x^{2} + x + 1 (- 3 x) + 1 \cdot 1)}$

Step 5.6.1.4

Multiply $- 3 x$ by $1$ .

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} - 3 x^{2} + x - 3 x + 1 \cdot 1)}$

Step 5.6.1.5

Multiply $1$ by $1$ .

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} - 3 x^{2} + x - 3 x + 1)}$

Step 5.6.2

Subtract $3 x$ from $x$ .

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (x^{2} - 3 x^{2} - 2 x + 1)}$

Step 5.7

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

$\frac{(x + 1) x \cdot 2 (8 x + 7)}{(x - 6) (x + 5) (- 2 x^{2} - 2 x + 1)}$

Step 6

Simplify with factoring out.

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Step 6.1

Move $2$ to the left of $(x + 1) x$ .

$\frac{2 \cdot ((x + 1) x) (8 x + 7)}{(x - 6) (x + 5) (- 2 x^{2} - 2 x + 1)}$

Step 6.2

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

$\frac{2 (x + 1) x (8 x + 7)}{(x - 6) (x + 5) (- (2 x^{2}) - 2 x + 1)}$

Step 6.3

Factor $- 1$ out of $- 2 x$ .

$\frac{2 (x + 1) x (8 x + 7)}{(x - 6) (x + 5) (- (2 x^{2}) - (2 x) + 1)}$

Step 6.4

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

$\frac{2 (x + 1) x (8 x + 7)}{(x - 6) (x + 5) (- (2 x^{2} + 2 x) + 1)}$

Step 6.5

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

$\frac{2 (x + 1) x (8 x + 7)}{(x - 6) (x + 5) (- (2 x^{2} + 2 x) - 1 (- 1))}$

Step 6.6

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

$\frac{2 (x + 1) x (8 x + 7)}{(x - 6) (x + 5) (- (2 x^{2} + 2 x - 1))}$

Step 6.7

Rewrite negatives.

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Step 6.7.1

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

$\frac{2 (x + 1) x (8 x + 7)}{(x - 6) (x + 5) (- 1 (2 x^{2} + 2 x - 1))}$

Step 6.7.2

Move the negative in front of the fraction.

$- \frac{2 (x + 1) x (8 x + 7)}{((x - 6) (x + 5)) (2 x^{2} + 2 x - 1)}$

Step 6.7.3

Reorder factors in $- \frac{2 (x + 1) x (8 x + 7)}{((x - 6) (x + 5)) (2 x^{2} + 2 x - 1)}$ .

$- \frac{2 x (x + 1) (8 x + 7)}{(x - 6) (x + 5) (2 x^{2} + 2 x - 1)}$