A polynomial determined by two values
Solution 1:
True. If $P(x)=a_nx^n+\cdots+a_1x+a_0$ then each of the coefficients are less then $b\equiv P(2)$. Each of these coefficients can then be read off from the base-b expansion of $P(b)=P(P(2))$.
Solution 2:
Look at $P(P(2))$ in base $P(2)$. The nth place is the coefficient of $x^n$.
Steve