Problem

Determine whether \(x^{2}+5x+6\) factors into nonconstant integer-coefficient linear factors. List the relevant factor pairs of \(6\), test their sums against \(5\), and give the factorization or a justified nonfactorability conclusion.

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Integer factorability requires one signed integer pair to satisfy both the product and sum conditions. We’ll list the relevant factor pairs, test their sums against the middle coefficient, expand any matching factorization, and use the discriminant as an independent rational-root check.

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The integers 2 and 3 have product 6 and sum 5, so x² + 5x + 6 = (x + 2)(x + 3). It factors over the integers; the discriminant 25 − 24 = 1 confirms this.

It factors because all of its coefficients are integers. Integer coefficients alone do not prove integer factorability.

The pair 2 and 3 shows that the expression is not factorable. In fact, that pair supplies the required product and sum.

The integers 1 and 6 have product 6, so the factorization is (x + 1)(x + 6). Their sum is 7, and the expansion has middle term 7x, not 5x.

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Determine whether \(x^{2}+5x+6\) factors into nonconstant integer-coefficient linear factors. List the relevant factor pairs of \(6\), test their sums against \(5\), and give the factorization or a justified nonfactorability conclusion.

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