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Adding 1m2−m−2+5mm2+3m+2\frac{1}{m^2-m-2} + \frac{5m}{m^2+3m+2}

Add two rational expressions with factorable trinomial denominators:

1m2−m−2+5mm2+3m+2\frac{1}{m^2 - m - 2} + \frac{5m}{m^2 + 3m + 2}

Step 1 — Find the LCD and rewrite each fraction. Factor the denominators: m2−m−2=(m−2)(m+1)m^2 - m - 2 = (m - 2)(m + 1) and m2+3m+2=(m+2)(m+1)m^2 + 3m + 2 = (m + 2)(m + 1). The LCD is (m−2)(m+1)(m+2)(m - 2)(m + 1)(m + 2). Multiply each fraction by its missing factor to rewrite them with the LCD:

1(m+2)(m−2)(m+1)(m+2)+5m(m−2)(m+2)(m+1)(m−2)\frac{1(m + 2)}{(m - 2)(m + 1)(m + 2)} + \frac{5m(m - 2)}{(m + 2)(m + 1)(m - 2)}

Distribute in each numerator: 1(m+2)=m+21(m + 2) = m + 2 and 5m(m−2)=5m2−10m5m(m - 2) = 5m^2 - 10m.

Step 2 — Add the numerators over the common denominator. Combine the numerators and collect like terms:

m+2+5m2−10m(m−2)(m+1)(m+2)=5m2−9m+2(m−2)(m+1)(m+2)\frac{m + 2 + 5m^2 - 10m}{(m - 2)(m + 1)(m + 2)} = \frac{5m^2 - 9m + 2}{(m - 2)(m + 1)(m + 2)}

Step 3 — Simplify, if possible. Check whether the numerator 5m2−9m+25m^2 - 9m + 2 can be factored. Look for two integers whose product is 5⋅2=105 \cdot 2 = 10 and whose sum is −9-9. No such integers exist, so the trinomial is prime. Since the numerator does not factor, it shares no common factors with the denominator. The expression is already in simplified form:

5m2−9m+2(m−2)(m+1)(m+2)\frac{5m^2 - 9m + 2}{(m - 2)(m + 1)(m + 2)}

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Updated 2026-04-30

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References

Tags

OpenStax

Intermediate Algebra @ OpenStax

Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax

Algebra

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