I have this assignment and I have to prove that $$ (b+aa^* b)+(b+aa^* b)(a+ba^* b)^* (a+ba^* b) = a^* b(a+ba^* a)b^* $$ How do I prove this?
What I have is this: $$\begin{align} \text{LHS} &= (b+aa^* b)(\varepsilon+(a+ba^* b)^* (a+ba^* b)) \\ &= (b+aa^* b)(a+ba^* b)^* \\ &= (\varepsilon+aa^* )b(a+ba^* b)^* \\ &= a^* b(a+ba^* b)^* \\ \end{align}$$
but I don't know how to proceed...
You won’t be able to prove it, because it’s false. The string $b$ matches $\color{blue}b+aa^*b$, so it matches the left-hand side. It does not match the right-hand side, however: the shortest string that matches $a^*\color{blue}b(\color{blue}a+ba^*a)b^*$ is $ba$.
Or, since your algebra is correct, just notice that $n$ matches your $a^*\color{blue}b(a+ba^*b)^*$ but not the original right-hand side.