The given expression is $8x^3 + 27$, which is a sum of cubes since:

$8x^3 = (2x)^3 \quad \text{and} \quad 27 = 3^3.$

The formula for factoring the sum of cubes $a^3 + b^3$ is:

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

Here:

• $a = 2x$
• $b = 3$

Applying the formula:

$8x^3 + 27 = (2x + 3)((2x)^2 - (2x)(3) + 3^2) = (2x + 3)(4x^2 - 6x + 9).$

Thus, the correct factorization is:

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

This matches the third option in the list: $(2x + 3)(4x^2 - 6x + 9)$.

Would you like further details or have any questions?

Here are some related questions for deeper understanding:

1. How do you factor differences of cubes?
2. What are the other forms of factorization (e.g., difference of squares)?
3. Can you use synthetic division to verify this factorization?
4. How does grouping work in factoring polynomials?
5. What are common mistakes to avoid when factoring sums of cubes?

Tip: Always remember the pattern for the sum and difference of cubes. It will save you time in factorization problems.