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Factoring ax² + bx + c by Grouping

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State

Factoring by grouping splits the middle term into two terms whose coefficients multiply to ac and add to b, then factors each half.

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Factor 2x² + 7x + 3. Here ac = 2 · 3 = 6 and b = 7; the pair 6 and 1 multiplies to 6 and adds to 7. Split the middle: 2x² + 6x + x + 3. Group and factor each half: 2x(x + 3) + 1(x + 3). The shared binomial factors out: (x + 3)(2x + 1). FOIL back: 2x² + x + 6x + 3 = 2x² + 7x + 3 — the original returns. The pair hunt is the whole search; the rest is the GCF move run twice.

Watch for

If no integer pair multiplies to ac and adds to b, the trinomial does not factor over the integers.

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