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$$x^2 - 507x + 2012 < 0$$ if $$x \in$$
$$x^2 - 507x + 2012 = (x - 4)(x - 503)$$, which is negative exactly when $$4 < x < 503$$. So the inequality holds for every $$x$$ in $$(4, 503)$$, and also for every $$x$$ in the smaller interval $$(10, 201)$$. It fails for some points of $$(1, 2012)$$ and of $$\mathbb{R} - (4, 503)$$. Hence options a and d are both correct.
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