OBM 2016 Problem 5

Consider the second-degree polynomial $P(x) = 4x^2+12x-3015$. Define the sequence of polynomials $P_1(x)=\frac{P(x)}{2016}$ and $P_{n+1}(x)=\frac{P(P_n(x))}{2016}$ for every integer $n \geq 1$.

[list='a'] []Show that exists a real number $r$ such that $P_n(r) < 0$ for every positive integer $n$. []Find how many integers $m$ are such that $P_n(m)<0$ for infinite positive integers $n$. [/list]