How to factor trinomials

Published October 2026.

Factoring ax² + bx + c into (px + q)(rx + s) is a search problem with a built-in test. Most students guess pairs and multiply them out to check, slow and error-prone. There is a faster way: check only the one number that matters.

The cross-check

Expanding (px + q)(rx + s) gives pr·x² + (ps + qr)·x + qs. So once pr = a and qs = c, only one condition remains: ps + qr = b, outer plus inner equals the middle coefficient. Test candidates against that single number instead of expanding everything.

The method, in order

1. Pull out the GCF first. 2x² + 8x + 6 = 2(x² + 4x + 3). Skipping this is the most common incomplete answer.
2. List factor pairs of a and of c (include negatives when c is positive and b is negative).
3. Cross-check each pair against ps + qr = b. The first pair that passes is your answer.
4. Expand once to verify. Thirty seconds that catches sign slips.

Worked example: 6x² + 11x + 3

Pairs of 6: (1,6), (2,3). Pairs of 3: (1,3). Try p=2, r=3, q=1, s=3: ps + qr = 2·3 + 1·3 = 9 ✗. Try q=3, s=1: ps + qr = 2·1 + 3·3 = 11 ✓. Answer: (2x + 3)(3x + 1).

When nothing works

If every pair fails the cross-check, the trinomial does not factor over integers, x² + x + 1 is a famous example. That is a complete answer, not a failure. The quadratic formula still finds the roots.

Practice with the factoring calculator, it shows the GCF step and the winning pair.

Special cases worth recognizing

Three patterns factor instantly once you spot them. Difference of squares: x² − 9 = (x + 3)(x − 3), no middle term, both signs. Perfect square trinomial: x² + 6x + 9 = (x + 3)², the middle term is twice the square root of the constant. Common factor hiding: 3x² + 12x + 12 = 3(x² + 4x + 4) = 3(x + 2)², always extract the GCF before looking for patterns. Recognizing these on sight turns a two-minute search into a ten-second answer.