If $\mathrm{x}=-5+4\mathrm{i}$ then ${\mathrm{x}}^4+9{\mathrm{x}}^3+35{\mathrm{x}}^2-\mathrm{x}+4=$

The given complex number is $\mathrm{x}=-5+4\mathrm{i}$

Take real parts on one side and then squaring on both sides 

$\begin{array}{rcl}\left(x+5\right)& =& 4i\\ {\left(x+5\right)}^2& =& -16\\ x^2+10x& =& -25-16\\ x^2+10x& =& -41\end{array}$

Consider the equation  ${\mathrm{x}}^4+9{\mathrm{x}}^3+35{\mathrm{x}}^2-\mathrm{x}+4$

$$\begin{gathered} {\mathrm{x}}^4+9{\mathrm{x}}^3+35{\mathrm{x}}^2-\mathrm{x}+4=x^2\left(x^2+9x+35\right)-x+4 \\ =x^2\left(-10x-41+9x+35\right)-x+4 \\ =x^2\left(-x-6\right)-x+4 \\ =\left(-10x-41\right)\left(-x-6\right)-x+4 \\ =10x^2+100x+250 \\ =10\left(-10x-41\right)+100x+250=-160 \end{gathered}$$