SMO 2012 Senior Round 1 Question 30

Let $P(x)$ be a polynomial of degree $34$ such that $P(k)=k(k+1)$ for all integers $k=0,1,2,\ldots,34$. Evaluate $42840P(35)$.



Editorial note: the displayed condition above transcribes the visible product in the supplied PDF. Its solution instead uses a quotient; see the review discussion.