You're ok up until the moment you distribute. I don't think the way you did that was valid. If you multiply back out again your factorization, you end up with

A(BB` + BC` + B`B` + B`C`) = A(0 + BC` + B` + B`C`) = A((B + B`)C` + B`) = A(C` + B`). This is not equal to A(B` + BC`). I don't think there's any way of simplifying any further. That is, A` + A(B` + BC`) is as far as you get.