$\displaystyle (x+y)^2+\left(\frac{8(x+y)}{x}\right)^2=225$
First square the second term:
$\displaystyle {\color{blue}(x+y)^{2}} + \frac{64{\color{blue}(x+y)^{2}}}{x^{2}} = 225$
Factor $\displaystyle (x+y)^{2}$ out:
$\displaystyle (x+y)^{2}\left( 1 + \frac{64}{x^{2}} \right) = 225$
Isolate $\displaystyle (x+y)^{2}$ and solving for y should be straightforward from here.