Question

If $z=-5+3i$ and the value of $z^4+9z^3+26z^2-14z+8$ is $k$, where $k\in R$, then the value of $-k$ is

Answer 1

$z=-5+3i\Rightarrow (z+5)=3i\Rightarrow (z+5{)}^2=-9\Rightarrow z^2+10z+34=0\cdots (1)$

Now using equation $\left(1\right)$, we get
$z^4+9z^3+26z^2-14z+8=z^2(z^2+10z+34)-z^3-8z^2-14z+8=0-z(z^2+10z+34)+2z^2+20z+8=2z^2+20z+8=2(z^2+10z+34)-60=-60\Rightarrow k=-60$
$\therefore -k=60$