The question reads:
Let $A\in M_{10}$, let $f(z)=z^{4}+11z^{3}-7z^{2}+5z+3$, and suppose $f(A)=0$. Prove that $A$ is invertible and find a polynomial $g$ of degree $3$ or less such that $A^{-1}=g(A)$.
The chapter the question comes from covers the Cayley-Hamilton Theorem and minimal polynomials, but I don't see how either of those topics help to show that the inverse exists given a random polynomial. I can find the polynomial $g$, but I don't see how to prove that $A$ is invertible.
From $f(A)=0$ we have $$-3I=A^4+11A^3-7A^2+5A = (A^3+11A^2-7A+5I)A.$$ Can you see what $A^{-1}$ must be?