The Perfect Square

One more shape worth recognizing on sight. x^2 + 6x + 9: the machine finds the pair 3 and 3, the SAME number twice, so the two parentheses match: (x + 3)(x + 3), written more honestly as (x + 3)^2.

That is a perfect square trinomial, and it has a signature you can check in a second: the constant is the square of half the middle coefficient. Half of 6 is 3, and 3^2 = 9. Signature confirmed, answer written down.

It is last section's (a + b)^2 = a^2 + 2ab + b^2 read backwards: middle 2ab, so half the middle is ab, and the constant is b^2. Square, double product, square, now recognized instead of built.