\( \frac{x+6}{x^{2}-2 x-8}+\frac{x+9}{x^{2}-5 x-14}=\square \) (Simplify your answer.)

To solve the equation \( \frac{x + 6}{x^{2} - 2x - 8} + \frac{x + 9}{x^{2} - 5x - 14} \), we first need to factor the denominators. 1. **Factoring the Denominators:** - \( x^{2} - 2x - 8 = (x - 4)(x + 2) \) - \( x^{2} - 5x - 14 = (x - 7)(x + 2) \) Now we can rewrite the equation: \[ \frac{x + 6}{(x - 4)(x + 2)} + \frac{x + 9}{(x - 7)(x + 2)} \] 2. **Finding a Common Denominator:** The common denominator here is \((x - 4)(x + 2)(x - 7)\). 3. **Rewrite Each Fraction:** - The first fraction becomes: \[ \frac{(x + 6)(x - 7)}{(x - 4)(x + 2)(x - 7)} \] - The second fraction becomes: \[ \frac{(x + 9)(x - 4)}{(x - 7)(x + 2)(x - 4)} \] 4. **Combine the Fractions:** Now, we can combine the fractions: \[ \frac{(x + 6)(x - 7) + (x + 9)(x - 4)}{(x - 4)(x + 2)(x - 7)} \] 5. **Simplifying the Numerator:** Expanding the numerators: - \( (x + 6)(x - 7) = x^2 - 7x + 6x - 42 = x^2 - x - 42 \) - \( (x + 9)(x - 4) = x^2 - 4x + 9x - 36 = x^2 + 5x - 36 \) Adding these together: \[ (x^2 - x - 42) + (x^2 + 5x - 36) = 2x^2 + 4x - 78 \] 6. **Final Expression:** The combined expression becomes: \[ \frac{2x^2 + 4x - 78}{(x - 4)(x + 2)(x - 7)} \] 7. **Further Simplification:** We can factor the numerator: \[ 2(x^2 + 2x - 39) \] Thus, our final simplified answer is: \[ \frac{2(x^2 + 2x - 39)}{(x - 4)(x + 2)(x - 7)} \] If we wish, we could leave it in this state or further factor \(x^2 + 2x - 39\) if possible, here it does not factor easily. So the simplified expression is: \[ \frac{2(x^2 + 2x - 39)}{(x - 4)(x + 2)(x - 7)} \]