The Fundamental Theorem of Algebra (Polynomial Form):
For any polynomial function,$f$, of degree $n$, there exists numbers $A$ and "roots", $r_1,r_2, ... , r_m$,  "powers", $p_1,p_2, ... , p_m$, and related linear functions, $f_j(x) =(x- r_j)^{p_j}$, with $ j = 1,2, ... , m$, and  a monic polynomial $g_f$ which has no real roots, so that $f$ can be expressed as a product :
$f(x) = A*f_1(x)*f_2(x)*...*f_m(x)*g_f(x)$
Furthermore: If $g_f \ne 1$ then there are "vertices," $(h_1,k_1), (h_2,k_2) ... (h_s,k_s)$  with $k_i > 0$ for all $i$ and powers $t_1,t_2, ... , t_s$, and related quadratic functions $q_{i}(x)=((x-h_i)^2 +k_i))^{t_i}$  so that $g_f$ can be expressed as a product::
$ g_f(x)=q_1(x)*q_2(x)*...*q_s(x)$