jmc

The polynomial $f(x)+cg(x)$ will have degree 3 exactly when the $x^4$ terms cancel and the $x^3$ terms do not. The $x^4$ term of $f(x)+cg(x)$ is $$5x^4+c(9x^4)=(5+9c)x^4.$$This is zero when $c=-5/9$.

If $c=-5/9$, the $x^3$ term is $$-4x^3+c(-6x^3)=(-4-6\cdot -5/9)x^3=-\frac{2}{3}{x}^3\ne 0.$$Therefore there is only one solution $c=\boxed{-\frac{5}{9}}$.