$A+A^2B+B=0$ implies $A^2+I$ invertible? Let $A$ and $B$ be two square matrices over a field such that $A+A^2B+B=0$.
Is it true that $A^2+I$ is always invertible ?

We have $A+(A^2+I)B=0$. We multiply by $A$:
$$A^2+(A^2+I)BA=0$$ 
We add $I$:
$$I+A^{2}+(A^2+I)BA=I=(A^2+I)(I+BA)$$
 Hence $A^2+I$ is invertible, and its inverse is $I+BA$. 
