How do you factorise $12^n - 4^n - 3^n +1$?

Question

So I split the $12$ into $(4)(3)$, but I'm not sure what to do next. I think a subtle hint may guide me - I bet its something about "adding zero" or turning the$1$ into something in terms of $4$ or $3$

Answer 1

$12^n-4^n-3^n+1=4^n(3^n-1)-3^n+1=(4^n-1)(3^n-1)$

and then you can use classical formulae to factor $(x^n-1)$.

Looking at the case n=1 you can easily see that this is the best you can do in the largest generality.

Answer 2

How about this: $$(3^n-1)(4^n-1)$$