Simplify (x^2+x-20)/(2x^2-32)*(2x^2-8x)/(2x^2-50)

$\frac{x^{2} + x - 20}{2 x^{2} - 32} \cdot \frac{2 x^{2} - 8 x}{2 x^{2} - 50}$

Step 1

Factor $x^{2} + x - 20$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $1$ .

$- 4, 5$

Write the factored form using these integers.

$\frac{(x - 4) (x + 5)}{2 x^{2} - 32} \cdot \frac{2 x^{2} - 8 x}{2 x^{2} - 50}$

Step 2

Simplify the denominator.

Factor $2$ out of $2 x^{2} - 32$ .

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

$\frac{(x - 4) (x + 5)}{2 (x^{2}) - 32} \cdot \frac{2 x^{2} - 8 x}{2 x^{2} - 50}$

Factor $2$ out of $- 32$ .

$\frac{(x - 4) (x + 5)}{2 x^{2} + 2 \cdot - 16} \cdot \frac{2 x^{2} - 8 x}{2 x^{2} - 50}$

Factor $2$ out of $2 x^{2} + 2 \cdot - 16$ .

$\frac{(x - 4) (x + 5)}{2 (x^{2} - 16)} \cdot \frac{2 x^{2} - 8 x}{2 x^{2} - 50}$

Rewrite $16$ as $4^{2}$ .

$\frac{(x - 4) (x + 5)}{2 (x^{2} - 4^{2})} \cdot \frac{2 x^{2} - 8 x}{2 x^{2} - 50}$

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 $b = 4$ .

$\frac{(x - 4) (x + 5)}{2 (x + 4) (x - 4)} \cdot \frac{2 x^{2} - 8 x}{2 x^{2} - 50}$

Step 3

Factor $2 x$ out of $2 x^{2} - 8 x$ .

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

$\frac{(x - 4) (x + 5)}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x) - 8 x}{2 x^{2} - 50}$

Factor $2 x$ out of $- 8 x$ .

$\frac{(x - 4) (x + 5)}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x) + 2 x (- 4)}{2 x^{2} - 50}$

Factor $2 x$ out of $2 x (x) + 2 x (- 4)$ .

$\frac{(x - 4) (x + 5)}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x - 4)}{2 x^{2} - 50}$

Step 4

Simplify the denominator.

Factor $2$ out of $2 x^{2} - 50$ .

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

$\frac{(x - 4) (x + 5)}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x - 4)}{2 (x^{2}) - 50}$

Factor $2$ out of $- 50$ .

$\frac{(x - 4) (x + 5)}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x - 4)}{2 x^{2} + 2 \cdot - 25}$

Factor $2$ out of $2 x^{2} + 2 \cdot - 25$ .

$\frac{(x - 4) (x + 5)}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x - 4)}{2 (x^{2} - 25)}$

Rewrite $25$ as $5^{2}$ .

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

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 $b = 5$ .

$\frac{(x - 4) (x + 5)}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x - 4)}{2 (x + 5) (x - 5)}$

Step 5

Simplify terms.

Cancel the common factor of $x + 5$ .

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

$\frac{(x + 5) (x - 4)}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x - 4)}{2 (x + 5) (x - 5)}$

Factor $x + 5$ out of $2 (x + 5) (x - 5)$ .

$\frac{(x + 5) (x - 4)}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x - 4)}{(x + 5) (2 (x - 5))}$

Cancel the common factor.

$\frac{(x + 5) (x - 4)}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x - 4)}{(x + 5) (2 (x - 5))}$

Rewrite the expression.

$\frac{x - 4}{2 (x + 4) (x - 4)} \cdot \frac{2 x (x - 4)}{2 (x - 5)}$

Cancel the common factor of $2 (x - 4)$ .

Factor $2 (x - 4)$ out of $2 (x + 4) (x - 4)$ .

$\frac{x - 4}{2 (x - 4) (x + 4)} \cdot \frac{2 x (x - 4)}{2 (x - 5)}$

Factor $2 (x - 4)$ out of $2 x (x - 4)$ .

$\frac{x - 4}{2 (x - 4) (x + 4)} \cdot \frac{2 (x - 4) (x)}{2 (x - 5)}$

Cancel the common factor.

$\frac{x - 4}{2 (x - 4) (x + 4)} \cdot \frac{2 (x - 4) x}{2 (x - 5)}$

Rewrite the expression.

$\frac{x - 4}{x + 4} \cdot \frac{x}{2 (x - 5)}$

Multiply $\frac{x - 4}{x + 4}$ by $\frac{x}{2 (x - 5)}$ .

$\frac{(x - 4) x}{(x + 4) (2 (x - 5))}$

Simplify the expression.

Move $2$ to the left of $x + 4$ .

$\frac{(x - 4) x}{2 \cdot (x + 4) (x - 5)}$

Reorder factors in $\frac{(x - 4) x}{2 (x + 4) (x - 5)}$ .

$\frac{x (x - 4)}{2 (x + 4) (x - 5)}$

Simplify (x^2+x-20)/(2x^2-32)*(2x^2-8x)/(2x^2-50)

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