I have a polynomial with only natural numbers as coefficients. Now I search for all such polynomials with the following additional property:

When I plug every natural number in it there is an infinite set of natural numbers not being derived.

I think the answer should be all polynomials of degrees greater or equal 2 and all linear polynomials except for x,x+1,x+2,x+3,...

But I don't know whether this is right and I can't prove it.