Prove that for every $n\times n$ matrices $A,B$: $$Tr((AB^2)A)=Tr(A^2B^2)$$
I need a solution that doesn't use expansion. One more question comes into my mind: given $A,B$ are square matrices. For which condition of $A,B$ we can conclude that $Tr(AB)=Tr(BA)$? Thanks in advance.