Remainder of the polynomial division, knowing other remainders Let $p(x)$ be a polynomial of 3rd degree.
We know that the division of $p(x)$ by $x-4$ gives us a remainder of 2 and divided by $x+2$ gives us the remainder of 1.
What's the remainder of $p(x)$ by $(x-4)(x+1)$?
I've used the remainder theorem but I don't seem to get anywhere... 

We can write 
$$p(x)=(x-4)(ax^2+bx+c)+2.$$
Since $p(-2)=1$, we have
$$1=(-6)(4a-2b+c)+2\quad\Rightarrow\quad c=-4a+2b+\frac 16.$$
So, we can write
$$p(x)=ax^3+(b-4a)x^2+\left(-4a-2b+\frac 16\right)x+16a-8b+\frac 43.$$
Thus, we have
$$p(x)=(x-4)(x+1)(ax+b-a)+\color{red}{\left(-3a+b+\frac 16\right)x+12a-4b+\frac 43}.$$
The answer is the red part. Here, we cannot eliminate $a,b$.
