(d) Solve for \( x \) : \( x+1=\sqrt{11+15 x+3 x^{2}-x^{3}} \)

Solve the equation \( x+1=\sqrt{11+15x+3x^{2}-x^{3}} \). Solve the equation by following steps: - step0: Solve for \(x\): \(x+1=\sqrt{11+15x+3x^{2}-x^{3}}\) - step1: Find the domain: \(x+1=\sqrt{11+15x+3x^{2}-x^{3}},x \in \left(-\infty,-\sqrt{15}+2\right]\cup \left[-1,\sqrt{15}+2\right]\) - step2: Swap the sides: \(\sqrt{11+15x+3x^{2}-x^{3}}=x+1\) - step3: Evaluate: \(\sqrt{11+15x+3x^{2}-x^{3}}=x+1,x+1\geq 0\) - step4: Evaluate: \(\sqrt{11+15x+3x^{2}-x^{3}}=x+1,x\geq -1\) - step5: Solve the equation: \(\begin{align}&x=5\\&x=-1\\&x=-2\end{align},x\geq -1\) - step6: Find the intersection: \(\begin{align}&x=5\\&x=-1\end{align}\) - step7: Check if the solution is in the defined range: \(\begin{align}&x=5\\&x=-1\end{align},x \in \left(-\infty,-\sqrt{15}+2\right]\cup \left[-1,\sqrt{15}+2\right]\) - step8: Find the intersection: \(\begin{align}&x=5\\&x=-1\end{align}\) - step9: Check the solution: \(\begin{align}&x=5\\&x=-1\end{align}\) - step10: Rewrite: \(x_{1}=-1,x_{2}=5\) The solutions to the equation \(x+1=\sqrt{11+15x+3x^{2}-x^{3}}\) are \(x=-1\) and \(x=5\).