Complex number solving

Question

Let $Z= -5+2i$ then find

$P(z) = z^4+11z^3+40z^2+39z+2046$

I know one way is that we find $z^4,z^3,z^2 $ and put values in it .which would be long, Is there any other way?

Answer 1

Evidently a contest problem

$$ \left( x^{4} + 11 x^{3} + 40 x^{2} + 39 x + 2046 \right) = \color{magenta}{ \left( x^{2} + 10 x + 29 \right) } \cdot \left( x^{2} + x + 1 \right) + \left( 2017 \right) $$

Answer 2

$(z+5)^2=(2i)^2\implies r(z)=z^2+10z+29=0$

Divide $P(z)\equiv?\pmod{r(z)}$