Final answer:
To solve the given expression involving fractions with quadratic denominators, factor the denominators, find the common denominator, and then simplify the expression or use the quadratic formula if a quadratic equation arises.
Step-by-step explanation:
To solve the expression (x²+4x+4)/(x²+3x+2)+(x+8)/(x²-5x-6), we must first factor the denominators. The quadratic expressions in the denominators factor as follows:
- (x²+4x+4) can be factored to (x+2)², as it is a perfect square.
- (x²+3x+2) factors to (x+1)(x+2).
- (x²-5x-6) factors to (x-6)(x+1).
Now rewrite the expression using these factors:
((x+2)²/(x+1)(x+2)) + ((x+8)/(x-6)(x+1))
Next, we find the common denominator, which is (x+1)(x+2)(x-6), and rewrite both fractions with this common denominator.
After we combine the terms, we simplify the numerator and check if any further simplification is possible. If there are common factors in the numerator and denominator, we simplify those as well. This process may lead us to a final simplified expression or a quadratic equation, depending on the resulting terms. To solve the quadratic equation, if one arises, we can use the quadratic formula if needed.