Answer
$\displaystyle \frac{y}{3}$
Work Step by Step
Dividing with $\displaystyle \frac{P}{Q}$ equals multiplying with the reciprocal, $\displaystyle \frac{Q}{P}.$
$ \displaystyle \frac{y^{2}-2y}{15}\div\frac{y-2}{5}=\frac{y^{2}-2y}{15}\cdot\frac{5}{y-2}\qquad$... factor what you can
$=\displaystyle \frac{y(y-2)}{15}\cdot\frac{5}{(y-2)}\qquad$... divide out the common factors
$=\displaystyle \frac{y(1)}{3}\cdot\frac{1}{(1)}$
= $\displaystyle \frac{y}{3}$