A Missing Middle

x^2 - 9 has no middle term, and that absence is a signature. Last section you expanded (x + 3)(x - 3) and watched the cross products cancel to nothing, leaving x^2 - 9. Factoring reads the signature backwards.

A difference of squares, x^2 minus a square number, splits into a sum and a difference of the same two things: x^2 - 9 = (x + 3)(x - 3). The 3 is the square root of the 9; the missing middle is the cancelled cross products.

Through the machine's eyes this is the pair 3 and -3: product -9, sum 0. The special shape is just the machine's b = 0 row, common enough and quick enough to deserve recognition on sight.