MathNet 2015-part-01 — 191 problems (EN–VI) Question 67

Let $P$ be a real polynomial of degree $2015$ and $Q$ a real quadratic polynomial. Could it be that the polynomial $P(Q(x))$ has precisely the roots
$$
-2014, -2013, \dots, -2, -1, 1, 2, \dots, 2014, 2015, 2016?
$$