Three. (20 points) Let $A$ and $B$ be points on the parabola $y=2 x^{2}+4 x-2$, with the origin being the midpoint of segment $AB$. Find the coordinates of points $A$ and $B$.

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Three, as shown in Figure 5, let $A(a, b)$. Since the origin is the midpoint of $A B$, $A$ and $B$ are symmetric with respect to the origin, i.e.,
$$
B(-a,-b) \text {. }
$$

Also, $A$ and $B$ are points on the parabola. Substituting them into the equation of the parabola, we get
$$
\left\{\begin{array}{l}
b=2 a^{2}+4 a-2, \\
-b=2 a^{2}-4 a-2 .
\end{array}\right.
$$

Solving these equations, we get $a=1, b=4$ or $a=-1, b=-4$.
Thus, $A(1,4)$ or $A(-1,-4), B(-1,-4)$ or $B(1, 4)$.