How to Factor by Grouping

Problem

x^3 + 2x^2 + 3x + 6

Answer

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

Four-term polynomials often factor two at a time. Group, factor each pair, and look for a shared bracket.

Step-by-step solution

Group the first two terms and the last two terms.
$(x^3 + 2x^2) + (3x + 6)$
Why: Pairing the terms makes each group share a common factor.
Factor each group.
$x^2(x + 2) + 3(x + 2)$
Why: The first group shares x^2 and the second shares 3.
Take out the common bracket.
$(x + 2)(x^2 + 3)$
Why: Both terms now hold the same bracket, (x + 2), so it factors out once.
Check by expanding.
$(x + 2)(x^2 + 3) = x^3 + 2x^2 + 3x + 6$
Why: The expansion matches the original four terms.

Sign errors in the second group: 3x + 6 gives +3.
Stopping at x^2(x + 2) + 3(x + 2) without factoring out the shared bracket.
Grouping terms that share nothing, such as (x^3 + 3x) + (2x^2 + 6), and then giving up.

Use the same method. Work each problem on paper, then open the answer to check.

$x^3 + 3x^2 + 4x + 12$

Show answer

$(x + 3)(x^2 + 4)$
$x^3 + 5x^2 + 2x + 10$

Show answer

$(x + 5)(x^2 + 2)$
$x^3 - 4x^2 + 3x - 12$

Show answer

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

Show answer

$(x + 1)(x^2 + 6)$
$x^3 - 2x^2 + 5x - 10$

Show answer

$(x - 2)(x^2 + 5)$