jmc

We can solve for $x$ and $y$, then plug them in to get our answer. However, there is a nicer method. Note that $$\begin{array}{rl}5x^2+8xy+5y^2& =(4x^2+4xy+y^2)+(x^2+4xy+4y^2)\\ & =(2x+y{)}^2+(x+2y{)}^2=4^2+5^2=\boxed{41}.\end{array}$$.