Generalisation of remainder thereom

Question

The Remainder theorem, says that when the polynomial $P(x)$ is divided by $x-a$, the remainder is $P(a)$.

What is the remainder when $P(x)$ is divided by $x^2 +ax +b$?

Answer 1

$$P(x) = (x^2 +ax +b) Q(x) + vx +t $$ if $z_1 , z_2 $ are solutions of $x^2 +ax +b =0 $ the the coefficients $v, t $ you can find from the linear system of equations $$vz_1 +t =P(z_1 ) ,vz_2 +t =P(z_2 )$$