Simplify ((x+4)/2)÷((x^2-16)/8)

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

To divide by a fraction, multiply by its reciprocal.

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

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the denominator.

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

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

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 $b = 4$ .

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

Cancel the common factor of $x + 4$ .

Cancel the common factor.

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

Rewrite the expression.

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