Choice C is the correct answer. This is a difference of squares: $(4x)^2 - 1^2$.

1. $a = 4x$, $b = 1$.
2. Formula: $(a - b)(a + b)$.
3. Result: $(4x - 1)(4x + 1)$.

Choice A is incorrect because it would have a middle term of $-8x$. Choice B is incorrect because it expands to $16x^2 + 15x - 1$. Choice D is incorrect because it expands to $16x^2 + 6x - 1$.