factoring by grouping

When a polynomial has four terms and no factor common to all of them, grouping is the trick that saves you. The idea is to split the four terms into two pairs, factor each pair on its own, and then hope the two pairs leave behind the same leftover bracket — which you can then pull out as a common factor. It is a two-stage version of taking out a common factor.

Take x^3 + 2x^2 + 3x + 6. Group as (x^3 + 2x^2) + (3x + 6). From the first pair factor out x^2 to get x^2(x + 2); from the second factor out 3 to get 3(x + 2). Now both pieces share the factor (x + 2), so the whole thing becomes (x + 2)(x^2 + 3). The matching bracket is the signal that the grouping worked.

Grouping does not always succeed: the way you pair the terms matters, and sometimes you must rearrange the order or factor a negative from one pair to make the leftover brackets agree. If no rearrangement makes the brackets match, the polynomial may simply not factor by this method.