Factor quadratic equation formula $ax^2+bx+c=(dx+e)(fx+g)$, and my goal is to find a formula to find $d,e,f,$ and $g$ when given $a,b,$ and $c$ and that also works for complex numbers
Without loss of generality the standard quadratic factors as ...
$$ ax^2+bx+c=a(x-d)(x-e) $$

where $d$ and $e$ are the two solutions you get from the quadratic formula)
By comparing the coefficients, we have
$$c=eg$$ 
$$a=df$$ 
$$b=dg+ef$$

There are three equations but four variables namely $d,e,f$ and $g$. So, you can not get a unique solution.