Simplify 1/(x^2+3x+2)-1/(x^2-2x-3)

Problem

1/(x2+3*x+2)−1/(x2−2*x−3)

Solution

1. Factor the denominators of both fractions.

x2+3*x+2=(x+1)*(x+2)

x2−2*x−3=(x+1)*(x−3)

2. Rewrite the expression using the factored denominators.

1/((x+1)*(x+2))−1/((x+1)*(x−3))

3. Find the least common denominator (LCD), which is (x+1)*(x+2)*(x−3)

LCD=(x+1)*(x+2)*(x−3)

4. Adjust each fraction to have the common denominator by multiplying the numerator and denominator by the missing factors.

(x−3)/((x+1)*(x+2)*(x−3))−(x+2)/((x+1)*(x−3)*(x+2))

5. Subtract the numerators over the common denominator.

((x−3)−(x+2))/((x+1)*(x+2)*(x−3))

6. Simplify the numerator by distributing the negative sign and combining like terms.

(x−3)−(x+2)=x−3−x−2=−5

7. State the final simplified expression.

(−5)/((x+1)*(x+2)*(x−3))

Final Answer

1/(x2+3*x+2)−1/(x2−2*x−3)=(−5)/((x+1)*(x+2)*(x−3))

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