Solve by factoring \[ x+1=-x^{2} \]

Solve the equation \( x+1=-x^{2} \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(x+1=-x^{2}\) - step1: Swap the sides: \(-x^{2}=x+1\) - step2: Move the expression to the left side: \(-x^{2}-x-1=0\) - step3: Multiply both sides: \(x^{2}+x+1=0\) - step4: Solve using the quadratic formula: \(x=\frac{-1\pm \sqrt{1^{2}-4}}{2}\) - step5: Simplify the expression: \(x=\frac{-1\pm \sqrt{-3}}{2}\) - step6: Simplify the expression: \(x=\frac{-1\pm \sqrt{3}\times i}{2}\) - step7: Separate into possible cases: \(\begin{align}&x=\frac{-1+\sqrt{3}\times i}{2}\\&x=\frac{-1-\sqrt{3}\times i}{2}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=-\frac{1}{2}+\frac{\sqrt{3}}{2}i\\&x=\frac{-1-\sqrt{3}\times i}{2}\end{align}\) - step9: Simplify the expression: \(\begin{align}&x=-\frac{1}{2}+\frac{\sqrt{3}}{2}i\\&x=-\frac{1}{2}-\frac{\sqrt{3}}{2}i\end{align}\) - step10: Rewrite: \(x_{1}=-\frac{1}{2}-\frac{\sqrt{3}}{2}i,x_{2}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i\) - step11: Remove the complex number(s): \(\textrm{No real solution}\) The equation \(x+1=-x^{2}\) does not have any real solutions. If you would like to find the complex solutions, please let me know.