jmc

Since $f(g(x))=g(x{)}^3-6g(x{)}^2+3g(x)-4$, it suffices to identify the constant terms of $g(x{)}^3$, $g(x{)}^2$, and $g(x)$. When expanding $g(x{)}^3$, we notice that the only way to obtain the constant term is to multiply the constant term with $g(x)$ by itself multiple (3) times: $(-2)\times (-2)\times (-2)=-8$. Similarly, the constant term of $g(x{)}^2$ is $(-2)\times (-2)=4$. The constant term in $g(x)$ is $-2$. Substituting gives $(-8)-6\cdot 4+3\cdot (-2)-4=-8-24-6-4=\boxed{-42}$.