jmc

We have $f(-4)=3(-4{)}^2-7=41$, so we seek $g(f(-4))=g(41)$. But what's $g(41)$? So, we turn to the other information we are given, $g(f(4))=9$. Since $f(4)=3(4{)}^2-7=41$, this equation gives us $g(41)=\boxed{9}$.