x² + 7x + 12
- 1.Two numbers with product 12 and sum 7: 3 and 4
(x + 3)(x + 4)
Expand back: x² + 4x + 3x + 12 = x² + 7x + 12 ✓.
Factoring, with the expand-back proof built in
Type a polynomial. You get the factors and the steps — then the calculator multiplies its own answer back out to prove it lands exactly on your expression. A factorization you can check is a factorization you can trust.
Scope: polynomials up to degree 2 in one variable, with rational factors. Higher-degree polynomials are not covered yet.
- ✓100% free — no paywall, ever
- ✓No sign-up, no app
- ✓Every answer verified before you see it
In short
A factoring calculator rewrites an expression as a product of simpler factors — for a trinomial x² + bx + c, the two brackets whose terms multiply back to the original. VerifySolve uses the GCF and AC method with every step shown, then proves the result by expanding the factors and recovering your exact starting expression.
The patterns worth memorising
Almost every school factoring problem is one of these four patterns in disguise. Spotting the pattern first is the whole skill — the arithmetic afterwards is easy.
Trinomial (a = 1)
AC method (a ≠ 1)
Difference of squares
Greatest common factor
Worked examples — each one expanded back
Read the “note” under each example: it is the expand-back check, the same verification the live solver performs on your problems.
x² − x − 20
- 1.Two numbers with product −20 and sum −1: −5 and 4
(x − 5)(x + 4)
Expand back: x² + 4x − 5x − 20 = x² − x − 20 ✓.
6x² + 11x + 3
- 1.AC method: ac = 18; two numbers with product 18 and sum 11: 9 and 2
- 2.Split and group: 6x² + 9x + 2x + 3 = 3x(2x + 3) + 1(2x + 3)
(3x + 1)(2x + 3)
Expand back: 6x² + 9x + 2x + 3 = 6x² + 11x + 3 ✓.
x² − 9
- 1.Recognise the difference of squares: x² − 3²
(x − 3)(x + 3)
Expand back: x² + 3x − 3x − 9 = x² − 9 ✓.
Common mistakes — and how to dodge them
Sign errors in the factor pair
For x² − x − 20 the pair must multiply to −20 (opposite signs) and add to −1 — so the bigger number is negative: −5 and 4. Always check both conditions; a pair that only satisfies one is the classic trap.
Skipping the GCF
Factor 2x² + 10x + 12 directly and you will wrestle with bigger numbers than needed. Pull the 2 out first — 2(x² + 5x + 6) — and the trinomial factors at a glance. GCF always comes first.
Never checking by expanding
Expanding takes ten seconds and catches every sign slip. If the expansion does not reproduce the original exactly, the factorization is wrong — no matter how confident it felt. Our solver never skips this step; neither should you.
Factoring Calculator — questions
How does factoring a trinomial work?
For x² + bx + c, find two numbers that multiply to c and add to b — those numbers become the constants in (x + p)(x + q). For ax² + bx + c with a ≠ 1, the AC method splits the middle term using two numbers that multiply to a·c and add to b, then factors by grouping. The calculator shows whichever route applies.
How can I be sure a factorization is right?
Expand it back. If (x + 3)(x + 4) multiplies out to your original x² + 7x + 12, it is correct — full stop. VerifySolve does this expansion check automatically and shows it as the verification on every factoring answer.
What if an expression cannot be factored?
Then it is prime over the rationals, and the calculator says so plainly. Some expressions factor only with irrational or complex numbers — the solver will tell you it has no rational factorization rather than inventing one.
Does it take out the greatest common factor first?
Yes — when every term shares a factor, the GCF comes out first (3x + 6 becomes 3(x + 2)), and any trinomial inside is then factored further if it can be.
Can it factor differences of squares?
Yes. x² − 9 is handled as the quadratic it is, giving (x − 3)(x + 3) — the classic difference-of-squares pattern.
Is the factoring calculator free?
Yes — free, no sign-up, no app, and the steps (plus the expand-back proof) are never paywalled.
Keep solving
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Any equation type, routed to the right method automatically.
A note on honest use: VerifySolve is built for learning. Use the steps to understand the method, then close the tab and try one on your own — that is where the grade actually comes from. If your teacher or school has a policy on calculators, follow it.