The problem asks us to factor the expression $64x^3 + 27$ using the sum of cubes formula.

1. Problem Description

The problem asks us to factor the expression $64x^3 + 27$ using the sum of cubes formula.

2. Solution Steps

The sum of cubes formula is:

$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

In the given expression $64x^3 + 27$, we can rewrite it as $(4x)^3 + (3)^3$. Thus, we have $a = 4x$ and $b = 3$.

Applying the sum of cubes formula, we get:

$(4x)^3 + (3)^3 = (4x + 3)((4x)^2 - (4x)(3) + (3)^2)$

Simplify the expression:

$(4x + 3)(16x^2 - 12x + 9)$

3. Final Answer

$(4x+3)(16x^2-12x+9)$