Polynomial with constant ratio

Question

What are all polynomials $P(x)$ with real coefficients such that for some $c>1$, for any integer $x$ there exists another integer $y$ with $P(y)=cP(x)$?

The monomials $P(x)=ax^n$ work because we can take $c=2^n$ and have $P(2x)=cP(x)$. If $P$ is not a monomial, then perhaps we can use an argument about the leading term dominating the remaining terms for large $x$, but that the remaining terms add some small value so that there is no $y$ such that $P(y)$ is exactly $cP(x)$.

Answer 1

Linear polynomials of the form $ax+ak$ work for any integer $k \gt 1$. We choose $c=k,$ and $y=cx+c^2-c$, so $P(y)=P(cx+c^2-c)=acx+ac^2-ac+ac=cP(x)$ I have not shown that there are not others.