Hunting the Pair

To find the two numbers, list the factor pairs of the constant $c$ and check which pair adds to $b$. To factor $x^{2}+8x+12$, list the pairs of positive whole numbers that multiply to 12: 1 and 12, 2 and 6, and 3 and 4. Their sums are 13, 8 and 7. You need the pair that adds to 8, so choose 2 and 6.

So $x^{2}+8x+12 = \left(x+2\right)\left(x+6\right)$. The list is short, because a number has only a few factor pairs. The list is also complete. If no pair on the list adds to $b$, then no pair of whole numbers works.

Start the list with 1 and $c$. Then try 2, then 3, and so on, until the two factors meet. Going in order means you do not miss a pair. Most lists have only two or three pairs.