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The expanded form of the given expression \((x + 2y)^3 - (2x - y)^3\) is:

\[
-7x^3 + 18x^2y + 6xy^2 + 9y^3
\]

Would you like more details on how this expansion was done, or have any questions?

Here are five related questions to further explore this topic:
1. How do you factor a cubic expression like the one in this problem?
2. Can you apply the binomial theorem to expand higher powers of sums and differences?
3. How would the expansion change if the terms inside the cubes were more complex?
4. How would this expression behave if specific values for \(x\) and \(y\) were substituted?
5. Can you generalize the difference of cubes formula for any polynomial expressions?

**Tip:** When expanding cubic expressions, consider applying the binomial theorem to simplify the process step by step!

This guide explains how to expand and simplify the expression (x + 2y)^3 - (2x - y)^3 using the cubic expansion and binomial theorem. Key algebraic concepts and formulas are covered. Ideal for high school students in Grades 9-12.

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