Factoring by Grouping

When to use factoring by grouping

Factoring by grouping works on polynomials with four terms. Split into two pairs, factor each pair separately, then extract the common binomial.

Step-by-step solution: Factor $2x^3 + 3x^2 + 4x + 6$

Step 1 — Group the first two and last two terms:

$(2x^3 + 3x^2) + (4x + 6)$

Step 2 — Factor out the GCF from each group:

$x^2(2x + 3) + 2(2x + 3)$

Step 3 — Both groups contain $(2x + 3)$. Factor it out:

$(2x + 3)(x^2 + 2)$

Step 4 — Check by expanding: $(2x + 3)(x^2 + 2) = 2x^3 + 4x + 3x^2 + 6 = 2x^3 + 3x^2 + 4x + 6$ ✓

The key insight

After factoring each group, the binomial factor inside the parentheses must be the same in both groups. If it isn't, try rearranging or grouping differently.

Common mistakes

• Wrong grouping. If your split doesn't produce a common binomial, try grouping terms 1&3 and 2&4 instead.
• Sign errors in the second group. When the third term is negative, factor out $-1$: $x(4x+3) - 2(4x+3) = (4x+3)(x-2)$.