Getting sum of products from products of sum

Question

I need to write the following Boolean expression in the form of sum of products

$F(A,B,C,D)= (A+B+C+D)(A'+B'+C+D')(A'+C)(A+D)(B+C+D)$

I just want to know how to deal with the missing letters.

Is $ (A'+C) = (A'+B+C+D) $ or is it $(A'+B'+C+D')$

In both the ways, a term already in the expression is created.

Answer 1

The "trick" to add the missing letters is that, e.g.

$(A' + C)$ is equivalent to $(A' + C + 0)$

and in turn :

$0$ is equivalent to $(DD')$.

Thus we can substitute for $(A' + C)$

$(A' + C + DD')$

and this is, by Distributivity :

$(A' + C + D)(A' + C + D')$.