Simplify (2x^3-32x)/(2x^3+16x^2+32x)

$\frac{2 x^{3} - 32 x}{2 x^{3} + 16 x^{2} + 32 x}$

Simplify the numerator.

Factor $2 x$ out of $2 x^{3} - 32 x$ .

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

$\frac{2 x (x^{2}) - 32 x}{2 x^{3} + 16 x^{2} + 32 x}$

Factor $2 x$ out of $- 32 x$ .

$\frac{2 x (x^{2}) + 2 x (- 16)}{2 x^{3} + 16 x^{2} + 32 x}$

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 16)$ .

$\frac{2 x (x^{2} - 16)}{2 x^{3} + 16 x^{2} + 32 x}$

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

$\frac{2 x (x^{2} - 4^{2})}{2 x^{3} + 16 x^{2} + 32 x}$

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 $b = 4$ .

$\frac{2 x (x + 4) (x - 4)}{2 x^{3} + 16 x^{2} + 32 x}$

Simplify the denominator.

Factor $2 x$ out of $2 x^{3} + 16 x^{2} + 32 x$ .

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

$\frac{2 x (x + 4) (x - 4)}{2 x (x^{2}) + 16 x^{2} + 32 x}$

Factor $2 x$ out of $16 x^{2}$ .

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

Factor $2 x$ out of $32 x$ .

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

Factor $2 x$ out of $2 x (x^{2}) + 2 x (8 x)$ .

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

Factor $2 x$ out of $2 x (x^{2} + 8 x) + 2 x (16)$ .

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

Factor using the perfect square rule.

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

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 4}{x + 4}$

Simplify (2x^3-32x)/(2x^3+16x^2+32x)

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