The expression given is $8x^3 + 27y^3$, and you are asked to factor it completely.

This is a sum of cubes. The general formula for factoring a sum of cubes is:

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

In this case, $8x^3$ can be written as $(2x)^3$, and $27y^3$ can be written as $(3y)^3$. So, we have:

$a = 2x \quad \text{and} \quad b = 3y$

Now, applying the sum of cubes formula:

$8x^3 + 27y^3 = (2x + 3y)((2x)^2 - (2x)(3y) + (3y)^2)$

Simplifying inside the second parentheses:

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

Thus, the factored form of $8x^3 + 27y^3$ is:

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

Would you like further details on this, or do you have any questions?

Here are 5 related questions:

1. How would you factor $x^3 + 8$, another sum of cubes?
2. What is the difference of cubes formula, and how does it compare to the sum of cubes?
3. How would you approach factoring $64x^3 + 125y^3$?
4. Why does the sum of cubes factor into a binomial and a trinomial?
5. Can the expression $x^3 + 1$ be factored in a similar way? If so, how?

Tip: The sum and difference of cubes always factor into a binomial and a trinomial. Make sure to memorize the formulas!