Find all pairs such that $x^2 y + x + y$ is divisible by $xy^2 + y + 7$ 
Find all pairs $(x, y)$ of positive integers such that $x^2 y + x + y$ is divisible by $xy^2 + y + 7$. If there are too many to write, write a generic form.

I was thinking of rewriting the divisibility in a simpler way such as $xy^2 + y + 7\mid x^2 y + x + y \implies xy^2 + y + 7\mid x^2y-xy^2+x-y$ but I am not sure if that helps. Is there an easier way?

Hint:
$$\begin{array}{ccc}xy^2+y+7&|&x^2y+x+y\\
xy^2+y+7&|&x^2y^2+xy+y^2\\
xy^2+y+7&|&y^2-7x
\end{array}$$
Compare $y^2-7x$ with $xy^2+y+7$
