Synthetic division by $ax^2+bx+c$. I know that synthetic division can be used in order to find quotient $q(x)$ and remainder $r(x)$ of a polynomial $p(x)$ when it is divided by some linear polynomial like $x-c$. Now, does exist some procedure (another than long division) in order to find $q(x)$ and $r(x)$ when the divisor is $ax^2+bx+c$? Suppose $b^2-4ac&lt;0$.
Assume your divisor is $x-d$ rather than $x-c$ since $c$ is already used in the formation of the original polynomial. You can write the dividend $y$ as:

$y = a(x-d)^2 + b(x-d) + 2adx - ad^2 + c + bd = a(x-d)^2 + b(x-d) + 2ad(x-d) + 2ad^2 -ad^2 + c + bd = a(x-d)^2  + (b+2ad)(x-d) + ad^2 + bd + c = (x-d)(a(x-d) + b+2ad) + ad^2+bd+c = (x-d)(ax + ad + b) + ad^2+bd+c \Rightarrow q(x) = ax+ad+b, r(x) = ad^2+bd+c$.
For $ax^2 + bx +c$ with $b^2 - 4ac $>0 find the roots of the polynomial and apply the syntectic division twice in a row .