Factorization Methods on $\mathbb{C,R,Q}$ and $\Bbb Z_2$. I need to factorize the polynomial $x^5 + 2x^4 -5x^3 -10x^2 + 6x +12$ on $\Bbb C$, $\Bbb R$, $\Bbb Q$, $\Bbb Z_2$ my manual does not provide any particular method. It has introduced only the fundamental theorem of algebra Ruffini's theorem etc.
In $\Bbb Q$, I managed to find a factorization by grouping:

Group 1:  -10x2-5x3
Group 2:  x5+2x4
Group 3:  6x+12 

Pull out from each group separately:

Group 1:   (x+2) • (-5x2)
Group 2:   (x+2) • (x4)
Group 3:   (x+2) • (6)

so
(x+2) • (x4-5x2+6)
and it is fairly easy to see that $(x^4-5x^2+6) = (x^2-2) (x^2-3)$
so a factorization in Q is $x^5 + 2x^4 -5x^3 -10x^2 + 6x +12 = (x+2) (x^2-2) (x^2-3)$
I've really no I idea on how to factorize in $\mathbb{C}$ and $\mathbb{R}$. Is there a link between the factorization in Q and those in C/R.?

Yes, there is a link between factorizations over $\Bbb Q$ repectively $\Bbb Z$ and over $\Bbb C$.
We can factorize $x^2-a$ further over $\Bbb C$, even though it is irreducible over $\Bbb Q$ for $a$ being a nonsquare, namely by
$$
x^2-a=(x-\sqrt{a})(x+\sqrt{a}).
$$
All other factorizations have been discussed except the one over the field $\Bbb F_2$. Since we have, over $\Bbb Z[x]$, that $f=(x+2)(x^2-2)(x^2-3)$, it
follows that in $\Bbb F_2[x]$ we have
$$
f=x^3(x^2-1)=x^3(x-1)^2.
$$
here we used "Freshman's Dream", i.e., that $(x+y)^p=x^p+y^p$ for $p=2$.
