Squares Inside

The special shapes hide inside GCF pulls too. 2x^2 - 50 is not a difference of squares as written, since 2 and 50 are in the way. But pull the 2: 2(x^2 - 25), and there is the shape: 2(x + 5)(x - 5).

This is why GCF-first is a reflex and not a suggestion: the signatures you learned to read, the missing middle, the squared half, only show once the shared factor is out of the frame.

Scan after every pull. A parenthesis holding x^2 minus a square, or a trinomial passing the half-squared check, still has a move left, and factored completely means taking it.