An algebra problem by Paul Ryan Longhas

For the equation $(3\sqrt{x} + 2)(x + 5) = 0$ to hold true, we require either one of two conditions to occur:

$x + 5 = 0$ (i),

$3\sqrt{x} + 2 = 0$ (ii)

Solving for x in (i) produces $x = -5$ , which in turn yields the complex number $2 + 3\sqrt{-5} = 2 + 3\sqrt{5}i$ for (ii). However, solving for x in (ii) produces $\sqrt{x} = -\frac{2}{3}$ , which is never valid for non-negative x. Also, the original equation is rendered invalid for all $x \in (0, -5) \cup (-5, \infty)$ and also when (ii) equals $2^{i}$ since a nonzero complex number will result under either of these conditions.

Hence, choice D is the best answer.