The problem is to factor the expression $8x^6y^{12} + 27$.

1. Problem Description

The problem is to factor the expression $8x^6y^{12} + 27$.

2. Solution Steps

The expression $8x^6y^{12} + 27$ is a sum of cubes. We can rewrite it as $(2x^2y^4)^3 + (3)^3$.

The sum of cubes formula is:

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

Let $a = 2x^2y^4$ and $b = 3$.

Then $a^3 = (2x^2y^4)^3 = 8x^6y^{12}$ and $b^3 = 3^3 = 27$.

Applying the sum of cubes formula, we have:

$8x^6y^{12} + 27 = (2x^2y^4 + 3)((2x^2y^4)^2 - (2x^2y^4)(3) + (3)^2)$

$= (2x^2y^4 + 3)(4x^4y^8 - 6x^2y^4 + 9)$

3. Final Answer

$(2x^2y^4 + 3)(4x^4y^8 - 6x^2y^4 + 9)$