Set of Integers Satisfying Quadratic Inequality

Given $f(x) = ax^2 + bx + c$ with discriminant $D = b^2 - 4ac < 0$. Since $D < 0$, $f(x)$ has no real roots, which means $f(x)$ always retains the same sign as $a$ for all real $x$.

• If $a > 0$, $f(x) > 0$ for all real $x$. Therefore, $f(m) < 0$ is never true for any integer $m$. Thus $S = \emptyset$ (empty set).
• If $a < 0$, $f(x) < 0$ for all real $x$. Therefore, $f(m) < 0$ is true for all integers $m$. Thus $S$ is the set of all integers.

Hence, $S$ must necessarily be either the empty set or the set of all integers.