Consider polynomials $p(x)$ with the following property, called $(†)$. $(†)$: If r is a root of $p(x)$, then $r^2 − 4$ is also a root of $p(x)$. We want to find every quadratic polynomial of the form $p(x) = x^2 +bx+c$ such that $p(x)$ has two distinct roots, has integer coefficients, and has property $(†)$. Prove that there are exactly two such polynomials and list them.
If $r^2-4$ is a root then it would imply $(r^2-4)^2-4$ is a root which would again imply $((r^2-4)^2-4)^2-4$ is a root and so on. But since we can have only two roots, we must have all the roots except $r$ and $r^2-4$ to be equal to either of them. Is my reasoning correct? How to explicitly find the equations?
$$p(x)=x^2- x-4$$ indeed solving the equation we get $$x_1=\frac{1}{2} \left(1-\sqrt{17}\right);\;x_2=\frac{1}{2} \left(\sqrt{17}+1\right)$$ $$\left(\frac{1}{2} \left(1-\sqrt{17}\right)\right)^2-4=\frac{1}{2} \left(1-\sqrt{17}\right)$$ $$\left(\frac{1}{2} \left(1+\sqrt{17}\right)\right)^2-4=\frac{1}{2} \left(\sqrt{17}+1\right)$$