I've come across this homework problem in an algebra course:

I have the first part more or less squared away after looking at the solutions book. However, the solution given for the second part confuses me. I can follow all of the logic until the final part, underlined:

My questions are:
(1) Where is the very strong iff coming from?
(2) What is the motivation for wanting an arbitrary square in b? Doesn't this approach leave the question unanswered, since it deals with only a subset of possible choices for b?