Solve for \( y \). \( 4+\frac{1}{y+1}=\frac{5}{y+4} \) If there is more than one solution, separate them with commas, If there is no solution, dick on "No solution". \( y=\square \)

To solve the equation \( 4+\frac{1}{y+1}=\frac{5}{y+4} \), we first eliminate the fractions by multiplying both sides by \( (y+1)(y+4) \): \[ 4(y+1)(y+4) + 1(y+4) = 5(y+1) \] Expanding this gives: \[ 4(y^2 + 5y + 4) + (y + 4) = 5y + 5 \] Putting it all together: \[ 4y^2 + 20y + 16 + y + 4 = 5y + 5 \] Now simplify: \[ 4y^2 + 21y + 20 = 5y + 5 \] Rearranging yields: \[ 4y^2 + 16y + 15 = 0 \] Next, we can factor or apply the quadratic formula. This factors neatly to: \[ (4y + 3)(y + 5) = 0 \] Setting each factor equal to zero gives us: \[ 4y + 3 = 0 \quad \Rightarrow \quad y = -\frac{3}{4} \] \[ y + 5 = 0 \quad \Rightarrow \quad y = -5 \] Thus, the solutions are: \[ y = -\frac{3}{4}, -5 \] Final answer: \( y = -\frac{3}{4}, -5 \)