If $x$ and $y$ are integers such that $5 \mid x^2 - 2xy - y$ and $5 \mid xy - 2y^2 - x$, prove that $5 \mid 2x^2 + y^2 + 2x + y$. 
Given that $x$ and $y$ are integers satisfying $5 \mid x^2 - 2xy - y$ and $5 \mid xy - 2y^2 - x$, prove that $5 \mid 2x^2 + y^2 + 2x + y$.

I have provided a (dumbfounding) solution down below if you want to check out. There should be simpler solutions, I believe so.

If $5\mid x^2-2xy-y$ and $5\mid xy-2y^2-x$ then $5$ also divides
$$(3y-1)(x^2-2xy-y)-(3x+2)(xy-2y^2-x)=2x^2+y^2+2x+y.$$
