I don't know how this can be simplified or solved, tried synthetic division.. no good.
Help appreciated
$\displaystyle \frac {x^4 +(2xy^2)^2 -3y}{(xy)^2} $
I don't know if you would consider this simpler, but
$\displaystyle \displaystyle \begin{align*} \frac{x^4 + (2xy^2)^2 - 3y}{(xy)^2} &= \frac{x^4 + 2x^2y^4 - 3y}{x^2y^2} \\ &= \frac{x^4}{x^2y^2} + \frac{2xy^2}{x^2y^2} - \frac{3y}{x^2y^2} \\ &= \frac{x^2}{y^2} + \frac{2}{x} - \frac{3}{x^2y} \end{align*}$