Solve $x^4+4x^3+5x^2+2x-2=0$, given that $-1+\sqrt{-1}$  is a root.

One root be $-1+i$  then the other root be $-1-i$

$\begin{array}{l}{x}^2-(-2)x+\left(2\right)=0\\ x^2+2x+2=0\end{array}$

Solve $x^2+2x-1$ then we will get remaining two roots

$\begin{array}{l}{x}^2+2x-1=0\\ x=\frac{-2\pm \sqrt{4+4}}{2}=\frac{-2\pm 2\sqrt{2}}{2}=-1\pm \sqrt{2}\end{array}$

$\therefore$ Roots of the given equation are $-1+i,-1-i,-1+\sqrt{2},-1-\sqrt{2}$