Guessing root of polynomials 
Given $p(x)=x^5+(1+2i)x^4-(1+3i)x^2+8+44i$ check with the Horner-scheme if $(-2-i)$ is a root of $p(x)$. 

First I have to guess a root, then proceed with the Horner-method and if i factorized it, i can say if $(-2-i)$ is a root or not, but how can i guess the first root, are there any tricks ?

$p(x) = 8 + 44i + x^2(-(1+3i) + x^2(1+2i + x))$
Now, let $x=-2-i$, thus, $x^2 = 4-1-4i = 3+4i$
$\begin{align}p(x) &= 8 +44i + (3+4i)(-(1+3i) + (3+4i)(-1+i)))\\
&=8+44i + (3+4i)(-(1+3i)  -3+3i-4i-4))\\
&=8+44i +(3+4i)(-1-3i-3+3i-4i-4)\\
&=8+44i + (3+4i)(-8 - 4i)\\
&=8+44i -24 -12i -32i +16\\
&=0
\end{align}
$
Thus $-2-i$ is a root.
