$T$ is an operator on $R^4$. And $T^3 + 3T^2 =4I$. Let $S= T^4 +3T^3 -4I$. Is $S$ invertible?
After a little calculation I found that $x(x+12)^2$ is an annihilating polynomial of $S$. Now the minimal polynomial divides the annihilating polynomial. Now if the minimal polynomial has zero root , then $0$ is an eigenvalue, thus $S$ isn't one-one, hence not invertible. But if $12$ is the only eigenvalue of $S$, then what? I can't go further.