I'm about to sound like a nerd here. when you factor and set equal to zero, the zero product property says that only one factor can equal zero. If you divide both sides by (a+b) instead of (a-b), you get (a-b) = 0 --> 1 - 1 = 0, and there's your zero factor that makes (a-b)(a+b) = 0 there.
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