Solve the equation. \( \frac{4}{w^{2}+2 w-8}+\frac{5}{w^{2}+w-6}=\frac{4}{w^{2}+7 w+12} \)

To solve the equation \[ \frac{4}{w^{2}+2 w-8}+\frac{5}{w^{2}+w-6}=\frac{4}{w^{2}+7 w+12}, \] we first factor the quadratic expressions in the denominators. 1. **Factoring**: - \( w^2 + 2w - 8 = (w + 4)(w - 2) \) - \( w^2 + w - 6 = (w + 3)(w - 2) \) - \( w^2 + 7w + 12 = (w + 3)(w + 4) \) So, substituting the factored forms, the equation becomes: \[ \frac{4}{(w + 4)(w - 2)} + \frac{5}{(w + 3)(w - 2)} = \frac{4}{(w + 3)(w + 4)}. \] 2. **Finding a common denominator**: The common denominator for the left-hand side is \( (w + 4)(w - 2)(w + 3) \). Rewrite each fraction: \[ \frac{4(w + 3)}{(w + 4)(w - 2)(w + 3)} + \frac{5(w + 4)}{(w + 3)(w - 2)(w + 4)} = \frac{4(w - 2)}{(w + 3)(w + 4)(w - 2)}. \] Hence, the equation simplifies to: \[ 4(w + 3) + 5(w + 4) = 4(w - 2). \] 3. **Expanding and combining like terms**: Expanding both sides gives us: \[ 4w + 12 + 5w + 20 = 4w - 8. \] Combining like terms yields: \[ (4w + 5w + 12 + 20) = 4w - 8 \implies 9w + 32 = 4w - 8. \] 4. **Solving for \( w \)**: Subtract \( 4w \) from both sides: \[ 9w - 4w + 32 = -8, \] which simplifies to: \[ 5w + 32 = -8. \] Now, subtract \( 32 \): \[ 5w = -8 - 32 \implies 5w = -40. \] Finally, divide by \( 5 \): \[ w = -8. \] 5. **Verification**: We need to check if \( w = -8 \) does not make any denominator zero: - \( w^2 + 2w - 8 = (-8)^2 + 2(-8) - 8 = 64 - 16 - 8 = 40 \) (not zero) - \( w^2 + w - 6 = (-8)^2 + (-8) - 6 = 64 - 8 - 6 = 50 \) (not zero) - \( w^2 + 7w + 12 = (-8)^2 + 7(-8) + 12 = 64 - 56 + 12 = 20 \) (not zero) Thus, the solution is \[ \boxed{-8}. \]