TSTST 2014 Problem 4

Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients, with $P \neq 0$, and let $d=\operatorname{deg} P$. Prove that there exist polynomials $A(x)$ and $B(x)$, not both zero, such that $\max \{\operatorname{deg} A, \operatorname{deg} B\} \leq d / 2$ and $P(x) \mid A(x)+Q(x) \cdot B(x)$.