To multiply the expressions \((x + 2y)(x^2 - xy + 3y)\), we will use the distributive property (also known as the FOIL method for binomials). Each term in the first expression \((x + 2y)\) will be multiplied by each term in the second expression \((x^2 - xy + 3y)\).

Let's expand step-by-step:

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

Now distribute:

1. **Distribute \(x\) to each term in the second expression:**

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

2. **Distribute \(2y\) to each term in the second expression:**

\[
2y(x^2) - 2y(xy) + 2y(3y) = 2x^2y - 2xy^2 + 6y^2
\]

Now, combine all the terms:

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

Next, combine like terms:

- Combine \( -x^2y \) and \( 2x^2y \):

\[
x^3 + ( -x^2y + 2x^2y) + 3xy - 2xy^2 + 6y^2 = x^3 + x^2y + 3xy - 2xy^2 + 6y^2
\]

So, the final expanded expression is:

\[
\boxed{x^3 + x^2y + 3xy - 2xy^2 + 6y^2}
\]

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. What is the distributive property and how is it applied in polynomial multiplication?
2. How do you identify like terms when combining expressions?
3. What are other methods to multiply polynomials, like binomials and trinomials?
4. Can you explain how to factor the result back into its original form?
5. What are common mistakes to watch out for when expanding polynomials?

**Tip:** When multiplying expressions, always ensure you multiply each term in one expression by every term in the other expression to avoid missing any terms.

Learn how to expand and multiply the polynomial expression (x + 2y)(x^2 - xy + 3y) step-by-step using the distributive property. This problem covers key algebra concepts such as polynomial multiplication, combining like terms, and is suitable for students in Grades 9-11.

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