Let $p(x)$ be a polynomial. When $p(x)$ is divided by $x-1$, it leaves the remainder $4$. When it is divided by $x-2$ it leaves the remainder $5$. Then find the remainder when $p(x)$ is divided by $x^2-3x+2$.

$p(1)=4$

$p(2)=5 $

$\Rightarrow p(x)=x+3.$

[/TEX]

I don't think I am right on guessing p(x)=x+3, If yes how can I divide it by a quadratic polynomial.