The Shape in Letters

The difference of squares, stated once for every case: a^2 - b^2 = (a + b)(a - b). Any two square things, any letters or numbers inside them: the difference of the squares is the sum times the difference.

You proved the right-to-left direction last section by expanding; the cross products ab and -ab cancel. The letters make the cancellation visible once and for all, with no particular numbers to distract.

This identity even does arithmetic: 41 * 39 is (40 + 1)(40 - 1) = 1600 - 1 = 1599. Recognize the shape in both directions and both chapters, multiplying and factoring, become one fact.