smc

Let the coordinates of $A$ be $(x_A,y_A)$. As $A$ lies on the parabola, we have $y_A=4x_A^2+7x_A-1$. As the origin is the midpoint of $AB$, the coordinates of $B$ are $(-x_A,-y_A)$. We need to choose $x_A$ so that $B$ will lie on the parabola as well. In other words, we need $-y_A=4(-x_A{)}^2+7(-x_A)-1$. Substituting for $y_A$, we get: $-4x_A^2-7x_A+1=4(-x_A{)}^2+7(-x_A)-1$. This simplifies to $8x_A^2-2=0$, which solves to $x_A=\pm 1/2$. Both roots lead to the same pair of points: $(1/2,7/2)$ and $(-1/2,-7/2)$. Their distance is $\sqrt{1^2+7^2}=\sqrt{50}=\boxed{5\sqrt{2}}$.