Solve for \( w \). \[ -\frac{1}{w+2}=-4+\frac{3}{w+3} \] If there is more than one solution, separate them with commas. If there is no solution, click on "No solution". \( w=-\frac{3}{2} \), \( [\square] \)

Solve the equation \( -\frac{1}{w+2}=-4+\frac{3}{w+3} \). Solve the equation by following steps: - step0: Solve for \(w\): \(-\frac{1}{w+2}=-4+\frac{3}{w+3}\) - step1: Find the domain: \(-\frac{1}{w+2}=-4+\frac{3}{w+3},w \in \left(-\infty,-3\right)\cup \left(-3,-2\right)\cup \left(-2,+\infty\right)\) - step2: Multiply both sides of the equation by LCD: \(-\frac{1}{w+2}\times \left(w+2\right)\left(w+3\right)=\left(-4+\frac{3}{w+3}\right)\left(w+2\right)\left(w+3\right)\) - step3: Simplify the equation: \(-w-3=-4w^{2}-17w-18\) - step4: Move the expression to the left side: \(-w-3-\left(-4w^{2}-17w-18\right)=0\) - step5: Subtract the terms: \(16w+15+4w^{2}=0\) - step6: Factor the expression: \(\left(3+2w\right)\left(5+2w\right)=0\) - step7: Separate into possible cases: \(\begin{align}&3+2w=0\\&5+2w=0\end{align}\) - step8: Solve the equation: \(\begin{align}&w=-\frac{3}{2}\\&w=-\frac{5}{2}\end{align}\) - step9: Check if the solution is in the defined range: \(\begin{align}&w=-\frac{3}{2}\\&w=-\frac{5}{2}\end{align},w \in \left(-\infty,-3\right)\cup \left(-3,-2\right)\cup \left(-2,+\infty\right)\) - step10: Find the intersection: \(\begin{align}&w=-\frac{3}{2}\\&w=-\frac{5}{2}\end{align}\) - step11: Rewrite: \(w_{1}=-\frac{5}{2},w_{2}=-\frac{3}{2}\) The solutions to the equation are \( w = -\frac{5}{2} \) and \( w = -\frac{3}{2} \). Therefore, the correct answer is \( w = -\frac{3}{2} \).