Points $A$ and $B$ lie on the parabola $y=2x^2+4x-2,$such that the origin is the mid point of the line segment $AB$ Points $A$ and $B$ lie on the parabola $y=2x^2+4x-2,$such that the origin is the mid point of the line segment $AB$.Find the length of the line segment $AB$

$y=2(x^2+2x-1)=2(x+1)^2-4\implies (y+4)=2(x+2)^2$ and let $x=t-2,y=2t^2-4$ be the parametric equation of the parabola.
I am stuck here.

Let the points be $A(x_1,2x_1^2+4x_1-2)$ and $B(x_2,2x_2^2+4x_2-2)$. Then:
$$\begin{cases}\frac12(x_1+x_2)=0\\ \frac12(2x_1^2+4x_1-2+2x_2^2+4x_2-2)=0\end{cases} \Rightarrow (x_1,x_2)=(\pm 1,\mp 1).$$
Hence: $A(1,4)$, $B(-1,-4)$ and $AB=\sqrt{(-1-1)^2+(-4-4)^2}=\sqrt{68}$.
