The Machine in Letters

The machine in one portable sentence: x^2 + (p+q)x + pq = (x + p)(x + q). Any pair p and q, any trinomial they build: the sum sits in the middle, the product at the end, and the pair drops into the parentheses.

Read left to right it is the expansion you proved last section with firsts, outers, inners, lasts. Read right to left it is this chapter's machine. One identity, both directions of travel.

Letters make the rule checkable: expand (x + p)(x + q) yourself and watch (p+q)x and pq appear. Every specific hunt you run is this sentence with numbers in place of p and q.