We are asked to factor the expression $27a^3 + 64b^3$. This is a sum of cubes.

1. Problem Description

We are asked to factor the expression $27a^3 + 64b^3$. This is a sum of cubes.

2. Solution Steps

We can recognize the given expression as a sum of cubes.

The general formula for the sum of cubes is:

$x^3 + y^3 = (x + y)(x^2 - xy + y^2)$

We can write $27a^3 + 64b^3$ as $(3a)^3 + (4b)^3$.

Here, $x = 3a$ and $y = 4b$.

Substituting these values into the sum of cubes formula, we get:

$(3a)^3 + (4b)^3 = (3a + 4b)((3a)^2 - (3a)(4b) + (4b)^2)$

$= (3a + 4b)(9a^2 - 12ab + 16b^2)$

3. Final Answer

$(3a + 4b)(9a^2 - 12ab + 16b^2)$