if $a'b+cd'=0$, then prove that $ab+c'(a'+d')=ab+bd+b'd'+a'c'd$

Question

Let $a'b+cd'=0$, then prove that $$ab+c'(a'+d')=ab+bd+b'd'+a'c'd$$

I would like to know how to solve this expression, not able to make any headway. I have tried canonical form expansion and reduction but the terms in the if condition does not match and also not able to generate the terms in the given expression.

Answer 1

A truth table shows that left-hand-side and righ-hand-side are in fact equivalent if the constraint is fulfilled (indicated by background color):