Permuted quadratic equation (Posted on 2021-03-23)

Prove that there doesn't exist a set of three different quadratic trinomials f(x), g(x), h(x) such that

the solutions of the equation f(x)=g(x) are 1 and 4, the solutions of the equation g(x)=h(x) are 2 and 5, and the solutions of the equation h(x)=f(x) are 3 and 6.