$(C + \bar{A})(AB + AB\bar{C})$ Simplifying Boolean

Question

Basically I have this terrible teacher that only gave us a ton of different examples using DeMorgran's Law, and now that one of these can't be solved using DeMorgan's Law, I am pretty lost on what to do. The whole question is below, the problem is that the only rule he has shown us is DeMorgan's Law...

Use the Boolean Logic rules provided to simplify the following expression. $(C + \bar{A})(AB + AB\bar{C})$.

Answer 1

\begin{align} (C+\overline{A})(AB+ABC')&=ABC+ABC\overline{C}+A\overline{A}B+\overline{A}AB\overline{C}\\&=ABC+AB\varnothing +\varnothing B+\varnothing B\overline{C}\\&=ABC\end{align}

Answer 2

$$(C+A')(AB+ABC')$$ $$=(C+A')AB(1+C')$$

As $1+C'=C$ in $GF(2)$,

$$=(C+A')ABC$$ $$=CABC+A'ABC$$

As $A'A=0$ and $CC=C$ in $GF(2)$,

$$=ABC$$