If –1 + i is a root of ${\mathrm{x}}^4 + 4{\mathrm{x}}^3 + 5{\mathrm{x}}^2 + 2\mathrm{x} + \mathrm{k} = 0$ then its real roots are

$\begin{array}{l}If-1+iisrootof{x}^4+4x^3+5x^2+2x+k=0\\ Then-1-iisalsoroot\\ \therefore TheEquationis{x}^2+2x+2=0\\ \therefore x^4+4x^3+5x^2+2x+kisdividedby{x}^2+2x+2\\ BysyntheticDivision\end{array}$

$\begin{array}{l}\therefore k+2=0,k=-2\\ \therefore x^4+4x^3+5x^2+2x-2=(x^2+2x+2)(x^2+2x-1)\\ x^2+2x-1rootsare\\ x=\frac{-2\pm \sqrt{4+4}}{2}\Rightarrow \frac{2\pm 2\sqrt{2}}{2}\Rightarrow 1\pm \sqrt{2}\end{array}$