Find all non-negative integer polynomials that satisfy : $P(1) = 8 $ ; $P(2) =2012 $ Find all non-negative integer polynomials that satisfy : $P(1) = 8 $ ;   $P(2) =2012 $
First, I set $Q(x) = P(x)+ax+b $ such that $1$ and $2$ are solutions of $ Q(x)$
$\Rightarrow 8+a+b = 0$  and  $2012 + 2a+b = 0$
$\Rightarrow a=-2004 ; b =1996 $
$\Rightarrow P(x) = (x-1)(x-2)R(x) +2004x-1996$
(Predicted result :$ P(x) = x^{10}+x^9+x^8+x^7+x^6+x^4+x^3+x^2 $)
But we have one more condition that the coefficients of $P(x)$ are non-negative integers, so I think I need to add another condition for $R(x)$ . But I don't have any more ideas. I hope to get help from everyone. Thanks very much !

So once you get that
$$P(x)=(x-1)(x-2)R(x)+2004x-1996$$
$$P(x)=(x^2-3x+2)R(x)+2004x-1996$$
We know that the coefficients of $R(x)$ must be integers. Let's say $R(x)$ is an integral polynomial of degree $n\geq 2$ (I'll leave the other cases to you, as they are pretty much the same process and simpler) so that
$$R(x)=a_0x^0+a_1x^1+\ldots a_nx^n=\sum_{k=0}^n a_kx^k$$
We have that
$$P(x)=(x^2-3x+2)R(x)+2004x-1996$$
$$P(x)=-1996+2004x+(x^2-3x+2)\sum_{k=0}^n a_kx^k$$
$$P(x)=-1996+2004x+\sum_{k=2}^na_{k-2}x^k+\sum_{k=1}^n -3a_{k-1}x^k+\sum_{k=0}^n 2a_kx^k$$
$$P(x)=(-1996+2a_0)+(2004-3a_0+2a_1)x+\sum_{k=2}^n a_{k-2}x^k+\sum_{k=2}^n -3a_{k-1}x^k+\sum_{k=2}^n 2a_kx^k$$
$$P(x)=(-1996+2a_0)+(2004-3a_0+2a_1)x+\sum_{k=2}^n \left((a_{k-2}-3a_{k-1}+2a_k)x^k\right)$$
So $R(x)$ is defined by any sequence of length $n$, $a_n$, that satisfies
$$\begin{cases} a_0\geq 998\\2a_1-3a_0\geq -2004\\ 2a_k-3a_{k-1}+a_{k-2}\geq 0~\forall k\geq 4\end{cases}$$
It's easy to see that there are infinite polynomials from there. Some more explicit, and simple constructions could be initial conditions that satisfy the first two inequalities, and then there immediately exists a sequence that satisfies the equality case $\forall k\geq 4$ (as long as you watch out for any parity conflicts).
Of course, another easy reason to see the existence of infinite polynomials is that we can easily make a valid polynomial for $R(x)$ of degree $n+1$ from a valid polynomial for $R(x)$ of degree $n$.
