Prove the irrationality of pi by contradiction

[etotheipi]

I think I've confused myself too , since Wikipedia gives a slightly different rundown of this part of the proof. Ultimately the summation still becomes an integer, but they seem to say there are some more constant terms:

I'll need to read over this again, because it's giving me a headache . I don't think we're too far off but there is perhaps a little bit left to this part of the proof.

 

[etotheipi]

I tried evaluating the 3rd derivative of $x^4(\pi - x)^4$ at $x=0$ and $x=\pi$ and I get zero on both counts, so I struggle to see what Wikipedia is trying to push here!