Choice A is the correct answer.

1. $a > 0$: Parabola opens upward.
2. $c < 0$: y-intercept is negative (below the x-axis).
3. Logic: If the graph crosses the y-axis below zero and eventually goes up to infinity on both sides, it MUST cross the x-axis twice (once on the way down, once on the way up).

Choice B is incorrect. Choice C is incorrect. Choice D is incorrect because the geometry guarantees two roots regardless of $b$.