If A and B are two non–zero n x n matrices such that A² + B = A²B then 

$\begin{array}{l}{\mathrm{A}}^2+\mathrm{B}={\mathrm{A}}^2\mathrm{B}\\ {\mathrm{A}}^2\mathrm{B}-{\mathrm{A}}^2-\mathrm{B}=0\\ \mathrm{B}\left({\mathrm{A}}^2-\mathrm{I}\right)-\left({\mathrm{A}}^2-\mathrm{I}\right)=\mathrm{I}\\ (\mathrm{B}-\mathrm{I})\left({\mathrm{A}}^2-\mathrm{I}\right)=\mathrm{I}\\ \Rightarrow {\mathrm{BA}}^2-\mathrm{B}-{\mathrm{A}}^2=0\Rightarrow {\mathrm{BA}}^2=\mathrm{B}+{\mathrm{A}}^2={\mathrm{A}}^2\mathrm{B}\\ \therefore {\mathrm{BA}}^2={\mathrm{A}}^2\mathrm{B}\end{array}$

OR

$\begin{array}{r}{A}^2+B=A^{2}B\\ \left(A^2-I\right)(B-I)=I\\ A^2+B=A^{2}B\\ A^2(B-I)=B\\ A^2=B(B-I{)}^{-1}\\ A^2=B\left(A^2-I\right)\\ A^2=B{A}^2-B\\ A^2+B=B{A}^2\\ A^{2}B=B{A}^2\end{array}$