a² − b² factors as (a − b)(a + b). If you see two squares with a minus and no extra middle term, that is the split. Factoring more generally is in factoring quadratics.
What it looks like
x² − 9 is (x − 3)(x + 3). 4y² − 25 is (2y − 5)(2y + 5). x² + 9 does not factor this way over the reals, because it is a sum, not a difference. If there is a middle term, it is an ordinary quadratic, not this shortcut.
After you factor, you can set each piece to zero. If the quadratic will not factor, use the quadratic formula. Completing the square is in completing the square.
The bottom line
Two squares, a minus, no linear term: write (a − b)(a + b) and stop FOILing from scratch. Polynomials around this pattern are in polynomials.
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