jmc

The polynomial $f(x)+cg(x)$ will have degree 2 exactly when the $x^3$ terms cancel and the $x^2$ terms do not. The $x^3$ term of $f(x)+cg(x)$ is $$4x^3+c(-6x^3)=(4-6c)x^3.$$This is zero when $c=4/6=2/3$.

If $c=2/3$, the $x^2$ term is $$3x^2+c(5x^2)=(3+5\cdot 2/3)x^2=\frac{19}{3}{x}^2\ne 0.$$Therefore there is only one solution $c=\boxed{\frac{2}{3}}$.