Consider the polynomial $ P(x+1)-P(x)-1$ it has degree $ \le 1$ so we get that we can not have more than $ n$ consecutive elements in the range of $ P$. This is reached when for example $ P(x)=x+\prod_{i=1}^n (x-i)$

Hi Albanian Eagle , What was your point to consider $ P(x+1) - P(x) - 1$ ? Why does the fact that its degree $ \leq 1$ deduce there are at most $ n$ consecutive elements? I think that we are to find the maximum number of integers $ a_1, a_2, ... a_k$ such that $ P(a_{i+1}) = P(a_i) + 1$ without the restriction of $ a_i$ to be consecutive integers.

Oh, right sorry for not explaining that part. Can you see why $ a_{i+1}-a_i|P(a_{i+1})-P(a_i)=1$