1Cademy - Simplifying $$\frac{x^2+5x+6}{x^2+8x+12}$$

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How to Simplify a Rational Expression

Example

Simplifying $\frac{x^2+5x+6}{x^2+8x+12}$

Simplify the rational expression $\frac{x^2+5x+6}{x^2+8x+12}$ by factoring the numerator and denominator as trinomials of the form $x^2 + bx + c$ and then canceling the common binomial factor.

Step 1 — Factor the numerator and denominator. The numerator $x^2 + 5x + 6$ requires two numbers whose product is $6$ and whose sum is $5$ : the pair $2$ and $3$ works, so $x^2 + 5x + 6 = (x + 2)(x + 3)$ . The denominator $x^2 + 8x + 12$ requires two numbers whose product is $12$ and whose sum is $8$ : the pair $2$ and $6$ works, so $x^2 + 8x + 12 = (x + 2)(x + 6)$ . The expression becomes:

$\frac{(x + 2)(x + 3)}{(x + 2)(x + 6)}$

Step 2 — Remove the common factor. The binomial $(x + 2)$ appears in both the numerator and the denominator, so it can be divided out:

$\frac{x + 3}{x + 6}$

The simplified result is $\frac{x + 3}{x + 6}$ . The values $x = -2$ and $x = -6$ must be excluded, because either value would make the original denominator equal zero. Unlike the earlier examples that used GCF factoring on binomials or monomials, this example requires factoring both the numerator and the denominator as trinomials before the shared polynomial factor becomes visible.

Updated 2026-04-21

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OpenStax Elementary Algebra

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