The expression $27a^3 + 64b^3$ can be factored using the sum of cubes formula, which is given by:

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

Here, we can identify $x = 3a$ and $y = 4b$ because:

$27a^3 = (3a)^3 \quad \text{and} \quad 64b^3 = (4b)^3$

So, the expression becomes:

$27a^3 + 64b^3 = (3a)^3 + (4b)^3$

Applying the sum of cubes formula:

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

Now, simplify the expression inside the second parentheses:

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

So, the fully factored form of $27a^3 + 64b^3$ is:

$27a^3 + 64b^3 = (3a + 4b)(9a^2 - 12ab + 16b^2)$

Would you like more details or have any questions?

Here are 5 related questions:

1. How would you factor the difference of cubes, such as $27a^3 - 64b^3$?
2. Can you apply the sum of cubes formula to a different set of terms, like $8x^3 + 27y^3$?
3. How would you factor a sum of squares, like $a^2 + b^2$?
4. What is the general formula for factoring $a^3 + b^3$ and $a^3 - b^3$?
5. How does the sum of cubes differ from the sum of squares in terms of factorization?

Tip: When dealing with cubic polynomials, always check if they fit the form of a sum or difference of cubes, as this allows for straightforward factoring.