Consider the polynomials  \[f(x)=1-12x+3x^2-4x^3+5x^4\] and \[g(x)=3-2x-6x^3+9x^4.\] Find $c$ such that the polynomial $f(x)+cg(x)$ has degree 3.
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   \[-4x^3+c(-6x^3)=(-4-6\cdot -5/9)x^3=-\frac{2}{3}x^3\neq0.\]Therefore there is only one solution $c=\boxed{-\frac{5}{9}}$.