Choice A is the correct answer.

1. Set $y = 0$: $x^2 + 4 = 0$.
2. Solve: $x^2 = -4$.
3. Since no real number squared equals a negative number, there are no real solutions.

Geometrically, this is a parabola shifted up 4 units, so it never touches the x-axis.

Choice B is incorrect because that happens if the vertex is on the axis. Choice C is incorrect because that happens if the vertex is below the axis (for upward opening). Choice D is incorrect because quadratics have at most 2.