The Deep Check

Multi-layer factorings get multi-layer checks, and the check is still multiplication. x^3 + 3x^2 + 2x = x(x + 1)(x + 2): multiply the parentheses first, (x + 1)(x + 2) = x^2 + 3x + 2, then the x back through, and the cubic returns.

Check from the inside out, one layer at a time, and a slip announces its own layer: wrong constant inside means the machine mis-hunted; wrong coefficients everywhere means the GCF was wrong.

This habit closes the chapter. Factor in layers, check in layers, and no expression, however tall, is anything but a stack of one-move problems.