2004 AMC 12B Problem 18

Points A and B are on the parabola y=4x^2+7x-1, and the origin is the midpoint of AB. What is the length of AB?

\mathrm{(A)}\ 2\sqrt5 \qquad \mathrm{(B)}\ 5+\frac{\sqrt2}{2} \qquad \mathrm{(C)}\ 5+\sqrt2 \qquad \mathrm{(D)}\ 7 \qquad \mathrm{(E)}\ 5\sqrt2

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.