Reduce ((2x^3)/(2y^2-7y-4))÷((6x^5)/(x^2-y^2))*(12-3y)/(x-y)

$\frac{2 x^{3}}{2 y^{2} - 7 y - 4} \div \frac{6 x^{5}}{x^{2} - y^{2}} \cdot \frac{12 - 3 y}{x - y}$

Factor by grouping.

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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 $a \cdot c = 2 \cdot - 4 = - 8$ and whose sum is $b = - 7$ .

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Factor $- 7$ out of $- 7 y$ .

$\frac{\frac{2 x^{3}}{2 y^{2} - 7 (y) - 4}}{\frac{6 x^{5}}{x^{2} - y^{2}}} \cdot \frac{12 - 3 y}{x - y}$

Rewrite $- 7$ as $1$ plus $- 8$

$\frac{\frac{2 x^{3}}{2 y^{2} + (1 - 8) y - 4}}{\frac{6 x^{5}}{x^{2} - y^{2}}} \cdot \frac{12 - 3 y}{x - y}$

Apply the distributive property.

$\frac{\frac{2 x^{3}}{2 y^{2} + 1 y - 8 y - 4}}{\frac{6 x^{5}}{x^{2} - y^{2}}} \cdot \frac{12 - 3 y}{x - y}$

Multiply $y$ by $1$ .

$\frac{\frac{2 x^{3}}{2 y^{2} + y - 8 y - 4}}{\frac{6 x^{5}}{x^{2} - y^{2}}} \cdot \frac{12 - 3 y}{x - y}$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$\frac{\frac{2 x^{3}}{(2 y^{2} + y) - 8 y - 4}}{\frac{6 x^{5}}{x^{2} - y^{2}}} \cdot \frac{12 - 3 y}{x - y}$

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

$\frac{\frac{2 x^{3}}{y (2 y + 1) - 4 (2 y + 1)}}{\frac{6 x^{5}}{x^{2} - y^{2}}} \cdot \frac{12 - 3 y}{x - y}$

Factor the polynomial by factoring out the greatest common factor, $2 y + 1$ .

$\frac{\frac{2 x^{3}}{(2 y + 1) (y - 4)}}{\frac{6 x^{5}}{x^{2} - y^{2}}} \cdot \frac{12 - 3 y}{x - y}$

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

$\frac{\frac{2 x^{3}}{(2 y + 1) (y - 4)}}{\frac{6 x^{5}}{(x + y) (x - y)}} \cdot \frac{12 - 3 y}{x - y}$

Factor $3$ out of $12 - 3 y$ .

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Factor $3$ out of $12$ .

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

Factor $3$ out of $- 3 y$ .

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

Factor $3$ out of $3 (4) + 3 (- y)$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2 x^{3}$ .

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Factor $2 x^{3}$ out of $6 x^{5}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{1}{(2 y + 1) (y - 4)}$ and $\frac{(x + y) (x - y)}{3 x^{2}}$ .

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

Cancel the common factor of $x - y$ .

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Factor $x - y$ out of $(x + y) (x - y)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{x + y}{(2 y + 1) (y - 4) x^{2}}$ and $4 - y$ .

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

Cancel the common factor of $4 - y$ and $y - 4$ .

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

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

Factor $- 1$ out of $- y$ .

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

Factor $- 1$ out of $- 1 (- 4) - (y)$ .

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x + y) \cdot - 1}{(2 y + 1) x^{2}}$

Simplify the expression.

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

$\frac{- 1 \cdot (x + y)}{(2 y + 1) x^{2}}$

Move the negative in front of the fraction.

$- \frac{x + y}{(2 y + 1) x^{2}}$

Reorder factors in $- \frac{x + y}{(2 y + 1) x^{2}}$ .

$- \frac{x + y}{x^{2} (2 y + 1)}$

Reduce ((2x^3)/(2y^2-7y-4))÷((6x^5)/(x^2-y^2))*(12-3y)/(x-y)