Choice C is the correct answer. The number of real solutions depends on the sign of the discriminant $\Delta = b^2 - 4ac$.

1. Calculate: $1^2 - 4(1)(5) = 1 - 20 = -19$.
2. Since $\Delta < 0$, there are no real solutions (the solutions are complex).

Choice A is incorrect because $\Delta$ is not positive. Choice B is incorrect because $\Delta$ is not zero. Choice D is incorrect because quadratic equations cannot have 3 solutions.