<s>[INST] unknown polynomial divided by $x^2(x-1)$, find the remainder. I took an exam today and there's a problem stuck in my head; I still can't figure out yet.
Here's the question (just the concept as I can't remember precisely). 
An unknown polynomial divided by $(x-1)^2$ leaves the remainder of $x + 3$ (not sure about the number) and when this polynomial is divided by $x^2 $, it leaves $2x + 4$ (again, not sure about the number). From the given conditions, if this polynomial is divided by $(x-1)x^2$, what would be the remainder?
The solution as far as I figured out is this: 
first, from the division of $(x-1)^2$, I got that $f(1) = 3$
in the same way from division of $x^2$, I got $f(0) = 4.$
I can write the polynomial as follows: 
$f(x) = (x-1)(x)(x) g(x) + ax^2 +bx +c$
$ax^2 + bx + c$ is the remainder.  And to find $a,b,c$, I can use the conditions above, so I got $c = 4$ by substituting $x = 0,$ and I got $a+b+4 = 3$ by substituting $x = 1.$
This leaves $a + b = -1,$ and I can't figure out how to continue; please help.
Edit : I made a mistake $f(1)$ should be equal to $4$ and $a+b+c = 4$
 [/INST] I am going to follow your approach. We have
$$f(x)=(x-1)^2p(x)+x+3$$
So, $f(1)=4$.
We also have that 
$$f(x)=x^2q(x)+2x+4$$
And we want to find $a,b,c$ in 
$$f(x)=(x-1)x^2g(x)+ax^2+bx+c$$
Plugin $x=1$ you get $4=a+b+c$.
Plugin $x=0$ doesn't give us enough information since this only gives us the remainder after dividing by $x$(instead of $x^2$).
So, instead of plugin, we look $f$ mod $x^2$. We have
$$f(x)=x^2[(x-1)g(x)+a]+(bx+c)$$
Since we are given that the remainder of dividing $f(x)$ by $x^2$ is $2x+4$, we conclude $bx+c=2x+4$, i.e.  $b=2$, $c=4$. And therefore, $a=-2$.
We conclude that the reminder of $f$ dividing by $ (x-1)x^2$ is $-2x^2+2x+4$.
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