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If $x=-5+2 \sqrt{-4}$, then the value of $x^4+9 x^3+35 x^2-x+4$ is
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The correct answer is:
$-160$
$x=-5+2 \sqrt{-4}$
$\begin{aligned} & x+5=4 i \\ & (x+5)^2=(4 i)^2 \\ & x^2+10 x+25=-16 \\ & x^2+10 x+41=0\end{aligned}$
Let $x^4+9 x^3+35 x^2-x+4=\left(x^2+10 x+41\right)(p(x))+\mathrm{R}$
$\begin{aligned} & x^2+10 x+41=0 \\ & x^2+9 x^3+35 x^2-x+4=\mathrm{R}\end{aligned}$
i.e. Remainder when $x^4+9 x^3+35 x^2-x+4$ is divided by $x^2+10 x+41$
$R=-160$
$x^4+9 x^3+35 x^2-x+4=-160$
$\begin{aligned} & x+5=4 i \\ & (x+5)^2=(4 i)^2 \\ & x^2+10 x+25=-16 \\ & x^2+10 x+41=0\end{aligned}$
Let $x^4+9 x^3+35 x^2-x+4=\left(x^2+10 x+41\right)(p(x))+\mathrm{R}$
$\begin{aligned} & x^2+10 x+41=0 \\ & x^2+9 x^3+35 x^2-x+4=\mathrm{R}\end{aligned}$
i.e. Remainder when $x^4+9 x^3+35 x^2-x+4$ is divided by $x^2+10 x+41$
$R=-160$
$x^4+9 x^3+35 x^2-x+4=-160$
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