factoring out a negative

Sometimes the obstacle to factoring is just a sign: the leading term is negative, or two brackets are almost identical but for their signs. Factoring out a negative means pulling -1 (often together with a common factor) to the front, which flips the sign of every term inside the parentheses. It is a cosmetic move that frequently uncovers structure that was hidden.

The mechanics: when you take -1 out of a sum, each term inside changes sign, because you are dividing each term by -1. So -3x + 6 becomes -3(x - 2), and -x^2 + 5x - 4 becomes -(x^2 - 5x + 4) = -(x - 1)(x - 4). Pulling out the negative also makes the leading coefficient positive, which is the conventional, tidier form.

This trick shines in factoring by grouping, where reversing a bracket lets two pieces match. The key identity to remember is that (a - b) = -(b - a): the brackets (x - 3) and (3 - x) are negatives of each other, and recognizing this lets you turn a mismatch into a common factor.