To factor the expression $27a^3 + 8b^3$, we recognize that this is a sum of cubes. The formula for factoring a sum of cubes is:

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

Here, $27a^3$ can be written as $(3a)^3$ and $8b^3$ as $(2b)^3$. So the expression becomes:

$(3a)^3 + (2b)^3$

Applying the sum of cubes formula, we get:

$(3a + 2b)\left((3a)^2 - (3a)(2b) + (2b)^2\right)$

Now simplify the second factor:

$(3a + 2b)\left(9a^2 - 6ab + 4b^2\right)$

Thus, the factored form of $27a^3 + 8b^3$ is:

$(3a + 2b)(9a^2 - 6ab + 4b^2)$

Would you like more details or have any questions?

Here are some related questions:

1. How would you factor $64x^3 + 1$?
2. Can you apply the difference of cubes formula to $125y^3 - 27z^3$?
3. How do you factor $a^3 + b^3$ in general?
4. How does factoring cubes relate to factoring squares?
5. What happens if you try to factor $27a^3 - 8b^3$?

Tip: For sum and difference of cubes, memorize their factorization formulas to easily break down these expressions.