$AB=A^2+A+I_n$ for real square matrices then $AB=BA$

Question

I know how to prove from $AB=A^2+A+I_n$ (with $A,B\in \mathcal{M}_n(\mathbb{R})$) that $AB=BA$ by showing that A is invertible. I would like to know if one can derive $AB=BA$ without using that $A$ is invertible.