Given a polynomial $P$ with integer coefficients taking certain values, find the largest number of integers that satisfy the equation $P(P(x)) = x^2$

Question

I do not know how to solve the following question:

Let us consider a polynomial with integers coefficients satisfying $P(-1)=-4$, $P(-3)= -40$ and $P(-5) = -156$. What is the largest possible number of integers $x$ satisfying $$P(P(x))=x^2?$$

As a first attempt, one can create a polynomial $P(x) = x^3 -x^2 +x-1$ that satisfies the constraints. In principle, there could be multiple polynomials satisfying the given constraints and it is not clear why the question should have a finite answer.