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 $g(6)$ .

Bonus: Generalize this.