Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Factor $2$ out of $- 32$ .

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

Step 1.1.3

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

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

Step 1.2

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

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

Step 1.3

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.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Factor $4$ out of $- 4$ .

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

Step 2.1.3

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

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

Step 2.2

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

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

Step 2.3

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.

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Step 3.1

Cancel the common factor of $2$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Cancel the common factor.

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

Step 3.1.4

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

Cancel the common factor of $x - 4$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x + 1$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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