Grouping All the Way

Sometimes the bracket left over after grouping factors again. Take x^3 + 2x^2 - 9x - 18. The first pair is x^2(x + 2). For the second pair, take out -9, not 9: -9x - 18 is -9(x + 2). Taking out the minus is what keeps the brackets matching.

Now take out the common factor: (x + 2)(x^2 - 9). But x^2 - 9 is a difference of squares, so it factors too:

x^3 + 2x^2 - 9x - 18 = (x + 2)(x + 3)(x - 3)

Factoring is finished only when no bracket factors any further. Each factor gives a root, so this cubic is zero at -2, -3 and 3.