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Factoring Trinomials of the Form $x^2 + bx + c$

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Factoring $x^2 + bx + c$ When $b$ Is Negative and $c$ Is Positive

When factoring a trinomial of the form $x^2 + bx + c$ where the middle coefficient $b$ is negative and the constant term $c$ is positive, both numbers $m$ and $n$ in the factored form $(x + m)(x + n)$ must be negative.

This follows from the sign rules for integer arithmetic. Recall that in the FOIL expansion $(x + m)(x + n) = x^2 + (m + n)x + mn$ :

The last term $c = mn$ is the product of $m$ and $n$ . A positive product requires factors with the same sign — both positive or both negative.
The middle coefficient $b = m + n$ is the sum of $m$ and $n$ . A negative sum requires the numbers to be negative (since two positive numbers always produce a positive sum).

Combining these two requirements — positive product and negative sum — the only possibility is that both $m$ and $n$ are negative. Therefore, when searching for factor pairs of $c$ , only negative pairs need to be tested. The factored form will use subtraction in both binomials: $(x - |m|)(x - |n|)$ .

Updated 2026-04-30

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