jmc

We see that $(x+2y{)}^2=(x^2+4y^2)+4xy=4^2=16$. We want to find $x^2+4y^2$ and are given $xy=-8$. So, $x^2+4y^2+4xy=x^2+4y^2+4(-8)=16$. It follows that $x^2+4y^2=\boxed{48}$.