Hunting the Pair

When the pair does not jump out, hunt it systematically: list the ways to multiply to c, then pick the one that sums to b. For x^2 + 8x + 12, the product 12 comes from 1 * 12, 2 * 6 or 3 * 4. The sums are 13, 8 and 7. The middle wants 8: the pair is 2 and 6.

So x^2 + 8x + 12 = (x + 2)(x + 6). The product list is short because c has only so many factor pairs, and it is complete: if no pair on the list sums to b, no whole-number pair exists at all.

Start the list at 1 times c and work inward; you will never miss a pair, and most hunts end after two or three lines. Speed comes later; completeness comes first.