Divide ((4x-x^2)/(x^3-64))÷(x/(x^2+4x+16))

$\frac{4 x - x^{2}}{x^{3} - 64} \div \frac{x}{x^{2} + 4 x + 16}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 x - x^{2}}{x^{3} - 64} \cdot \frac{x^{2} + 4 x + 16}{x}$

Step 2

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

Factor $x$ out of $4 x$ .

$\frac{x \cdot 4 - x^{2}}{x^{3} - 64} \cdot \frac{x^{2} + 4 x + 16}{x}$

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

$\frac{x \cdot 4 + x (- x)}{x^{3} - 64} \cdot \frac{x^{2} + 4 x + 16}{x}$

Factor $x$ out of $x \cdot 4 + x (- x)$ .

$\frac{x (4 - x)}{x^{3} - 64} \cdot \frac{x^{2} + 4 x + 16}{x}$

Step 3

Simplify the denominator.

Rewrite $64$ as $4^{3}$ .

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

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 4$ .

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

Simplify.

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

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

Raise $4$ to the power of $2$ .

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

Step 4

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Cancel the common factor of $x^{2} + 4 x + 16$ .

Factor $x^{2} + 4 x + 16$ out of $(x - 4) (x^{2} + 4 x + 16)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Cancel the common factor of $4 - x$ and $x - 4$ .

Rewrite $4$ as $- 1 (- 4)$ .

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

Factor $- 1$ out of $- x$ .

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

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

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

Reorder terms.

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

Cancel the common factor.

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

Divide $- 1$ by $1$ .

$- 1$

Divide ((4x-x^2)/(x^3-64))÷(x/(x^2+4x+16))

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