I need to find the characteristic polynomial $P_A$. And I know that $A^2+A+5I=0$, and $A$ is square matrix.
I don't know how to proceed. I think found that $A = -5(A+I)^{-1}$ but have no idea about the rest.
Update: $A$ is real and it is an $n \times n$ matrix.
If $A$ is a real matrix, then $P$ is a power of $x^2+x+5$, because the latter is irreducible over the reals.
If $A$ is a complex matrix, then all you can say is that the irreducible factors of $P$ are among the factors of $x^2+x+5$.