Can anyone show me how to find the 2 points that it intersects by hand (calculations)? I tried it by doing arctan x = πx^4 but got confused.
You aren't getting confused, you are faced with an impossible situation. You can't solve that equation in general. We can, however, pick out x = 0 as a solution fairly easily. There may well be a clever way to ferret out the other solution, but I doubt it.