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The problem asks to factor the cubic polynomial \( 8x^3 - 27 \). This expression is a **difference of cubes** since:

\[
8x^3 = (2x)^3 \quad \text{and} \quad 27 = 3^3
\]

The general formula for factoring a difference of cubes is:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Here:
- \( a = 2x \)
- \( b = 3 \)

Applying the formula:

\[
8x^3 - 27 = (2x - 3)\left((2x)^2 + (2x)(3) + 3^2\right)
\]
\[
= (2x - 3)(4x^2 + 6x + 9)
\]

Thus, the correct factored form is:

\[
(2x - 3)(4x^2 + 6x + 9)
\]

The correct answer from the given options is **b**.

Would you like further details or have any questions? Here are some related questions to explore:

1. What is the formula for factoring the sum of cubes?
2. How do you recognize the difference between a difference of squares and a difference of cubes?
3. What are some other examples of factoring using the difference of cubes formula?
4. How can you verify that a factored form is correct by expanding it?
5. How does factoring help in solving cubic equations?

**Tip**: When factoring, always check if the expression is a special case like a difference or sum of cubes.

Factor the following cubic polynomial: 8x^3 - 27

Learn how to factor the cubic polynomial 8x^3 - 27 using the difference of cubes formula. This problem covers key algebraic concepts, including factoring cubic equations. Suitable for students in Grades 9-12.

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