Points $A$ and $B$ are on parabola $y=3x^2-5x-3$, and the origin is the midpoint of $\overline{AB}$. Find the square of the length of $\overline{AB}$.
Since the origin is the midpoint of $\overline{AB}$, I can write the coordinates of $A$ and $B$ as $(x_1,y_1)$ and $(-x_1,-y_1)$, respectively. I also know that $A$ and $B$ are on the parabola, so they must satisfy the equation $y=3x^2-5x-3$. Substituting $(x_1,y_1)$ and $(-x_1,-y_1)$ into this equation, I get two equations: \begin{align*} y_1 &= 3x_1^2-5x_1-3 \\ -y_1 &= 3x_1^2+5x_1-3 \end{align*} Adding these two equations, I get $0=6x_1^2-6$, or $x_1^2=1$. This means that $x_1=\pm 1$. Without loss of generality, I can assume that $x_1=1$, since the other case will give the same answer. Now I can plug in $x_1=1$ into either equation to find $y_1$. I choose the first one and get $y_1=3-5-3=-5$. So the coordinates of $A$ and $B$ are $(1,-5)$ and $(-1,5)$, respectively. To find the square of the length of $\overline{AB}$, I can use the distance formula and square it: \begin{align*} \left(\overline{AB}\right)^2 &= \left(\sqrt{(-1-1)^2+(5-(-5))^2}\right)^2 \\ &= \left(\sqrt{4+100}\right)^2 \\ &= 4+100 \\ &= 104 \end{align*}

