Answer
$\displaystyle \frac{(x-2)^{2}}{x}$
Work Step by Step
Dividing with $\displaystyle \frac{P}{Q}$ equals multiplying with the reciprocal, $\displaystyle \frac{Q}{P}.$
$ \displaystyle \frac{x^{2}-4}{x}\div\frac{x+2}{x-2}=\frac{x^{2}-4}{x}\cdot\frac{x-2}{x+2}\qquad$... factor what you can
$=\displaystyle \frac{(x-2)(x+2)}{x}\cdot\frac{x-2}{x+2}\qquad$... divide out the common factors
$=\displaystyle \frac{(x-2)\cdot 1}{x}\cdot\frac{(x-2)}{1}\qquad$
= $\displaystyle \frac{(x-2)^{2}}{x}$