Find the roots of a complex square polynomial I'm stuck with this exercise and would welcome some help. I've been asked to:


  Find the roots in $\mathbb{C}$ and factorize in $\mathbb{C[x]}$ the polynomial $x^2 + (3+2i)x + 5 + i$


I tried many things. The first attempt was to see if I could factorize it just by looking at it. Then, I tried the formula $x=\frac{-b \pm w}{2a}$, where $w^2= b^2-4ac$

I did that and got $w^2= 8i + 15$ and then wrote $w= c+di$, since it's a complex number, and wrote a system of equations:

$a^2-b^2= -15$

$2ab= 8$

$a^2 + b^2= 289$

With that system I get that $a=\pm\sqrt{137}$, with that information I get $b$ and then go back to my previous equation with $x$, but that is already far from the answer, that is: $x_1=-1+i$, $x_2=-2-3i$

There's something obviously wrong in my reasoning. Could someone help me spot it?
Thanks a lot.
You should have got
$$
b^2-4ac=(3+2i)^2-4(5+i)=9+12i-4-20-4i=8i-15
$$
Now, setting $8i-15=(r+is)^2$ you get
$$
\begin{cases}
r^2-s^2=-15 \\[4px]
2rs=8 \\[4px]
r^2+s^2=\sqrt{8^2+15^2}=\sqrt{289}=17
\end{cases}
$$
Thus $2r^2=2$ and $r=\pm1$, together with $s=4/r$, so the square roots are $w_1=1+4i$ and $w_2=-1-4i$.

You can compute the roots as
$$
\frac{-b+w_1}{2}
\qquad\qquad
\frac{-b+w_2}{2}
$$
Notice $-b$, not $-b^2$.
Just use the standard formula for solving a quadratic equation.

It works for complex coefficients the same as real.