Consider the polynomial $f(x) = (g(x))^2 + (x^2 + 1)^2$. If $g(x) = 0$ holds for all $x \in S$, then $f(x)$ is a perfect square. If $g(x) > (x^2 + 1)^2$ holds for all $x \in \mathbb{Z} \setminus S$, then $(g(x))^2 < f(x) < (g(x) + 1)^2$, that is to say $f(x)$ is not a perfect square. Thus we may take $g(x) = M_S \prod_{s \in S} (x-s)^{10}$, where $M_S$ is a sufficiently large constant such that $g(x) > (x^2+1)^2$ holds for all $x \in \mathbb{Z} \setminus S$. Since the degree of $g$ is greater than 4, $M_S$ exists. $\square$