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

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

To divide by a fraction, multiply by its reciprocal.

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

Simplify the denominator.

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

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

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 $b = 4$ .

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

Simplify terms.

Cancel the common factor of $x - 4$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 4}{3 x} \cdot \frac{x + 4}{x + 4}$

Multiply $\frac{x - 4}{3 x}$ by $\frac{x + 4}{x + 4}$ .

$\frac{(x - 4) (x + 4)}{3 x (x + 4)}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{(x - 4) (x + 4)}{3 x (x + 4)}$

Rewrite the expression.

$\frac{x - 4}{3 x}$