Simplify (2x^2-32)/(x-4)*(x+1)/(4x^2-4)

$\frac{2 x^{2} - 32}{x - 4} \cdot \frac{x + 1}{4 x^{2} - 4}$

Step 1

Simplify the numerator.

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

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

$\frac{2 (x^{2}) - 32}{x - 4} \cdot \frac{x + 1}{4 x^{2} - 4}$

Factor $2$ out of $- 32$ .

$\frac{2 x^{2} + 2 \cdot - 16}{x - 4} \cdot \frac{x + 1}{4 x^{2} - 4}$

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

$\frac{2 (x^{2} - 16)}{x - 4} \cdot \frac{x + 1}{4 x^{2} - 4}$

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

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

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 $b = 4$ .

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

Step 2

Simplify the denominator.

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

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

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

Factor $4$ out of $- 4$ .

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

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

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

Rewrite $1$ as $1^{2}$ .

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

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 $b = 1$ .

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

Step 3

Simplify terms.

Cancel the common factor of $2$ .

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

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

Factor $2$ out of $4 (x + 1) (x - 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{(x + 4) (x - 4)}{x - 4}$ by $\frac{x + 1}{2 (x + 1) (x - 1)}$ .

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

Cancel the common factor of $x - 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (2x^2-32)/(x-4)*(x+1)/(4x^2-4)

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