I've already vaguely given you the answer

Algebra Level 5

Let \(g(x) \) denote a quintic \(\big(5^\text{th}\)-degree\(\big)\) polynomial such that \(g(n) = 2^n\) for \(n = 1,2,3,4,5\) and \(g(x) \) has a constant term of \(5!\).

Find \(g(6) \).