The expression \((x + 2y)(x^2 - xy + y^2)\) can be expanded using the distributive property (also known as the FOIL method in this case).

Let's expand it step by step:

\[
(x + 2y)(x^2 - xy + y^2)
\]

Distribute \(x\) to each term in the second polynomial:

\[
x \cdot (x^2 - xy + y^2) = x^3 - x^2y + xy^2
\]

Next, distribute \(2y\) to each term in the second polynomial:

\[
2y \cdot (x^2 - xy + y^2) = 2yx^2 - 2y^2x + 2y^3
\]

Now, combine all these results:

\[
x^3 - x^2y + xy^2 + 2x^2y - 2xy^2 + 2y^3
\]

Finally, combine like terms:

\[
x^3 + ( - x^2y + 2x^2y) + (xy^2 - 2xy^2) + 2y^3
\]

\[
x^3 + x^2y - xy^2 + 2y^3
\]

So, the expanded form of \((x + 2y)(x^2 - xy + y^2)\) is:

\[
x^3 + x^2y - xy^2 + 2y^3
\]

Would you like more details or have any other questions?

Here are five related questions to deepen your understanding:

1. How would the expression change if the signs in the second polynomial were different?
2. Can you identify the type of polynomial each factor is in the original expression?
3. What would the expansion look like if \(x = 1\) and \(y = -1\)?
4. How would you factor the expanded expression back into its original form?
5. What are the roots of the polynomial \((x + 2y)(x^2 - xy + y^2) = 0\)?

**Tip:** When expanding binomials, carefully distribute each term and keep track of like terms to simplify the expression efficiently.

This problem focuses on expanding and simplifying the expression (x + 2y)(x^2 - xy + y^2) using the distributive property. The detailed step-by-step solution covers the expansion of polynomials and combining like terms. Ideal for students in Grades 10-12 studying algebraic expressions.

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