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

1. Problem Description

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

2. Solution Steps

We recognize that the given expression is a sum of cubes.

The formula for the sum of cubes is:

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

We can rewrite the given expression as:

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

Here, $a = 2x^2y^4$ and $b = 3$. Applying the sum of cubes formula:

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

Now, we simplify the second factor:

$(2x^2y^4)^2 = 4x^4y^8$.

$(2x^2y^4)(3) = 6x^2y^4$.

$(3)^2 = 9$.

So the expression becomes:

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

3. Final Answer

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