Let $A \in M_n(\mathbb R)$ such that $$A^2+A+5I_n = 0$$ Find the characteristic polynomial of $A$.
I tried two different approaches and got stuck on both. I am wondering if I was even headed in the right direction.
First attempt:
Let $g(x) = x^2+x+5$. Using the datum we have $g(A) = 0$, thus implying through Cayley Hamilton theorem that $g(x) \setminus f_A(x)$.
Therefore we have, $f_A(x) = (x^2+x+5)h(x)$,
where $h(x)$ is some monic polynomial s.t deg($h$) = $n-2$.
This being the point I got stuck.
Second attempt:
By datum $A^2+A+5I=0 \implies A = -(A^2+5I) \implies f_A=f_{-(A^2+5I)}$
Therefore we have $f_A = det(xI+A^2+5I) = det((x+5)I+A^2)$.
This being the point I got stuck again unfortunately.
Hoping to get some direction and tips about what I've got and what I need.