Choice A is the correct answer.

1. Discriminant: $(-2)^2 - 4(3)(-2) = 4 + 24 = 28$.
2. Formula: $x = \frac{2 \pm \sqrt{28}}{6}$.
3. Simplify: $\sqrt{28} = 2\sqrt{7}$.
4. $x = \frac{2 \pm 2\sqrt{7}}{6} = \frac{2(1 \pm \sqrt{7})}{6} = \frac{1 \pm \sqrt{7}}{3}$.

Choice B is incorrect because it is the unsimplified form. While numerically correct, simplified forms are standard. Choice C is incorrect because the denominator is wrong ($2a=6$). Choice D is incorrect because of a sign error ($-b = 2$).