Intersecting lines through $t \rightarrow (t,t^2,t^3)$ curve.

Question

Can one find 4 distinct points on the above curve in $\mathbb R^3$, such that a line going through the first and third point intersects with the one passing through the other two?

Answer 1

Suppose the $4$ points lie on an affine plane $W$ (exercise: why is this equivalent to your problem?)

Then, $W=\{(x,y,z)\,:\,ax+by+cz+d=0\}$

But this means that $at^3+bt^2+ct+d=0$ has four distinct solutions.