Let $f(x)$ be a polynomial with integer coefficients. Suppose that there exist distinct integers $a_1,a_2,a_3,a_4,$ such that $f(a_1)=f(a_2)=f(a_3)=f(a_4)=3.$ Then show that there does not exist any integer $b$ with $f(b)=14.$
We have $f(x)=3+(x-a_1)(x-a_2)(x-a_3)(x-a_4)g(x)$ for some polynomial $g$ with integer coefficients. But then $(b-a_1)(b-a_2)(b-a_3)(b-a_4)g(b)=11$. Use the primality of $11$ to get a contradiction.
Hint: Consider the polynomial $g(x) = f(x) - 3$. Then $g$ has four known linear factors $(x - a_1), (x - a_2), (x - a_3), (x - a_4)$. What can these four factors evaluate to if $f(b) = 14$?