problem: Consider the polynomials  \[f(x)=4x^3+3x^2+2x+1\]and \[g(x)=3-4x+5x^2-6x^3.\]Find $c$ such that the polynomial $f(x)+cg(x)$ has degree 2.
solution: 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  \[3x^2+c(5x^2)=(3+5\cdot 2/3)x^2=\frac{19}{3}x^2\neq0.\]Therefore there is only one solution $c=\boxed{\frac{2}{3}}$.