Suppose you can factorize:
[tex]ax^2+bx+c=a(x-r_1)(x-r_2)[/tex]
Then by expanding the right side, you have
[tex]ax^2+bx+c=a(x^2-(r_1+r_2)x+r_1r_2)=ax^2-a(r_1+r_2)x+ar_1r_2[/tex]
which means in this case, you have to have
[tex]\begin{cases}6=-2(r_1+r_2)\\-8=2r_1r_2\end{cases}\implies\begin{cases}r_1+r_2=-3\\r_1r_2=-4\end{cases}[/tex]
