boolean algebra x=A.B.C+A'C
ABC + A'C

C (AB + A')

C (B + A') or BC + A'C


is correct.
Assuming '' is legitimate.  


A'' = NOT NOT A = A
First two terms are an XOR.  Multiply them out.

Then look for common terms and minimize.

Here's a link to boolean rules.  Try and repost.