Simplify (x^4+2x^3)/(4x^4+4x^3-8x^2)*(2x^2-2)/(x+1)

$\frac{x^{4} + 2 x^{3}}{4 x^{4} + 4 x^{3} - 8 x^{2}} \cdot \frac{2 x^{2} - 2}{x + 1}$

Step 1

Factor $x^{3}$ out of $x^{4} + 2 x^{3}$ .

Factor $x^{3}$ out of $x^{4}$ .

$\frac{x^{3} x + 2 x^{3}}{4 x^{4} + 4 x^{3} - 8 x^{2}} \cdot \frac{2 x^{2} - 2}{x + 1}$

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

$\frac{x^{3} x + x^{3} \cdot 2}{4 x^{4} + 4 x^{3} - 8 x^{2}} \cdot \frac{2 x^{2} - 2}{x + 1}$

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot 2$ .

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

Step 2

Simplify the denominator.

Factor $4 x^{2}$ out of $4 x^{4} + 4 x^{3} - 8 x^{2}$ .

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

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

Factor $4 x^{2}$ out of $4 x^{3}$ .

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

Factor $4 x^{2}$ out of $- 8 x^{2}$ .

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

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

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

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

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

Factor $x^{2} + x - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

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

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

Step 3

Simplify the numerator.

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

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

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

Factor $2$ out of $- 2$ .

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

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

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

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

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

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 $b = 1$ .

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

Step 4

Simplify terms.

Cancel the common factor of $2 (x - 1)$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{x^{3} (x + 2)}{2 x^{2} (x + 2)}$ by $\frac{x + 1}{x + 1}$ .

$\frac{x^{3} (x + 2) (x + 1)}{2 x^{2} (x + 2) (x + 1)}$

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x}{2}$

Simplify (x^4+2x^3)/(4x^4+4x^3-8x^2)*(2x^2-2)/(x+1)

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