Divide ((x+4)/(x^2+2))÷((x^2-2)/(x-8))

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Multiply $\frac{x + 4}{x^{2} + 2}$ by $\frac{x - 8}{x^{2} - 2}$ .

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

Step 3

Expand $(x + 4) (x - 8)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 8$ to the left of $x$ .

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

Multiply $4$ by $- 8$ .

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

Add $- 8 x$ and $4 x$ .

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

Step 5

Split the fraction $\frac{x^{2} - 4 x - 32}{(x^{2} + 2) (x^{2} - 2)}$ into two fractions.

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

Step 6

Split the fraction $\frac{x^{2} - 4 x}{(x^{2} + 2) (x^{2} - 2)}$ into two fractions.

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Move the negative in front of the fraction.

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