jmc

We see that $(x+y{)}^2=(x^2+y^2)+2xy=1$. We want to find $x^2+y^2$ and are given $xy=-4$. So, $x^2+y^2+2xy=x^2+y^2+2(-4)=1$. It follows that $x^2+y^2=\boxed{9}$.