Answer
$= \displaystyle \frac{(x-2)^{2}}{x},\qquad x\neq 0,-2,2$
Work Step by Step
Factor what we can. Recognize a difference of squares.
$\displaystyle \frac{x^{2}-4}{x}\div\frac{x+2}{x-2}=\frac{(x-2)(x+2)}{x}\div\frac{x+2}{x-2}$
Division with $\displaystyle \frac{a}{b}$ equals multiplication with $\displaystyle \frac{b}{a}.$
(neither a or b can be 0, we exclude $-2$ and $2$)
$=\displaystyle \frac{(x-2)(x+2)}{x}\cdot\frac{(x-2)}{(x+2)}$
...exclude any additional values that yield 0 in the denominator:
$x\neq 0,-2,2$
... cancel common factors
$= \displaystyle \frac{(x-2)^{2}}{x},\qquad x\neq 0,-2,2$
