[robinhouston]

Hmm, no. Look again. It's using the fact that (x^2 - y^2) = (x - y)(x + y), which is the case, and a common enough identity (“the difference of two squares“) that it's used without remark here.

The problem really is that you can't divide by zero, even in an algebraic expression.

A simpler example of this phenomenon (which blew my mind when I first encountered it) occurs with the equation x = x^2. If you divide by x, you get x = 1, which is a solution to the equation, but where did the other solution x = 0 go??

Whenever you divide an equation by an algebraic expression, you need to consider the possibility of that expression being zero and treat it as a special case. So in the case of x = x^2, you can reason as follows: maybe x = 0, in which case … what … ah yes, that's a solution! Or maybe x ≠ 0, in which case we can divide by it and get x = 1. That doesn't contradict the assumption x ≠ 0, so it's okay, and x = 1 is the other solution.