Question

Find the value of $x^4+9x^4+35x^4+4$ for $x=-5+2\sqrt{-4}$

Answer 1

$x^4+6x^3+35x^2+4=0$

Given that $x=-5+2\sqrt{-4}$

$x+5=2\times 2i$

$\Rightarrow x+5=4i$

Squaring both sides, we have

${(x+5)}^2={\left(4i\right)}^2$

$x^2+25+10x=16i^2$

$x^2+10x+25=-16[\because i^2=-1]$

$x^2+10x+41=0$

Now dividing $x^4+6x^3+35x^2+4$ by $x^2+10x+41$, we have

$(x^4+6x^3+35x^2+4)=(x^2+10x+41)(x^2-x+4)+(-160)$

Hence the value of $x^4+6x^3+35x^2+4=0$ for $x=-5+2\sqrt{-4}$ is $-160$.

Hence the required answer is $-160$.