Solve for \( u \). \[ 3+\frac{1}{u+6}=-\frac{5}{(u+1)(u+6)} \] If there is more than one solution, separate them with commas. If there is no solution, click on "No solution". \( u=\square \square \)

To solve the equation \[ 3+\frac{1}{u+6}=-\frac{5}{(u+1)(u+6)}, \] we'll first eliminate the fractions by multiplying both sides by \( (u+1)(u+6) \): \[ 3(u+1)(u+6) + (u+1) = -5. \] This expands to: \[ 3(u^2 + 7u + 6) + (u + 1) = -5. \] Distributing the 3 yields: \[ 3u^2 + 21u + 18 + u + 1 = -5. \] Combine the like terms: \[ 3u^2 + 22u + 19 = -5. \] Now, add 5 to both sides: \[ 3u^2 + 22u + 24 = 0. \] Next, we can use the quadratic formula: \[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-22 \pm \sqrt{22^2 - 4 \cdot 3 \cdot 24}}{2 \cdot 3}. \] Calculating the discriminant: \[ 22^2 = 484, \] and \[ 4 \cdot 3 \cdot 24 = 288. \] So the discriminant is: \[ 484 - 288 = 196. \] Now, taking the square root gives: \[ \sqrt{196} = 14. \] Now substituting back into the formula: \[ u = \frac{-22 \pm 14}{6}. \] Calculating the two possibilities for \( u \): 1. For \( u = \frac{-22 + 14}{6} = \frac{-8}{6} = -\frac{4}{3} \). 2. For \( u = \frac{-22 - 14}{6} = \frac{-36}{6} = -6 \). Thus, the solutions are: \[ u = -\frac{4}{3}, -6. \] Therefore, the final answer is: \( u = -\frac{4}{3}, -6 \)