reverse FOIL

FOIL is the routine for multiplying two binomials — First, Outer, Inner, Last. Reverse FOIL is doing that movie backward: you are handed the expanded trinomial and asked to recover the two binomials it came from. It is essentially trinomial factoring framed as 'un-multiplying,' and it leans on understanding how the pieces of FOIL produced each term.

The logic: in (px + m)(qx + n), the First terms multiply to give the x^2 coefficient (p·q), the Last terms multiply to give the constant (m·n), and the Outer plus Inner give the middle coefficient (p·n + q·m). To reverse it, you search for blanks p, q, m, n that hit all three targets at once, testing candidate factor pairs of the first and last coefficients until the middle term comes out right.

For trinomials with leading coefficient 1 this is quick, but when the leading coefficient is larger the trial-and-error can balloon, which is exactly why the AC method exists as a more systematic alternative. Reverse FOIL rewards practice: experienced solvers spot the right pair almost by feel.