To expand the expression \((x - 2y)(3x + y)\), you can use the distributive property (also known as the FOIL method for binomials). Here’s the step-by-step process:

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

1. **First**: Multiply the first terms from each binomial.
   \[
   x \times 3x = 3x^2
   \]

2. **Outer**: Multiply the outer terms.
   \[
   x \times y = xy
   \]

3. **Inner**: Multiply the inner terms.
   \[
   -2y \times 3x = -6xy
   \]

4. **Last**: Multiply the last terms from each binomial.
   \[
   -2y \times y = -2y^2
   \]

Now, combine all these results:
\[
3x^2 + xy - 6xy - 2y^2
\]

Combine like terms (the \(xy\) terms):
\[
3x^2 - 5xy - 2y^2
\]

So, the expanded expression is:
\[
3x^2 - 5xy - 2y^2
\]

Would you like more details on this process or have any questions?

Here are five related questions:

1. How do you factor quadratic expressions like \(3x^2 - 5xy - 2y^2\)?
2. What is the difference between expanding and factoring in algebra?
3. How do you apply the distributive property to more complex expressions?
4. Can you simplify expressions involving higher powers, such as \(x^3\) or \(y^3\)?
5. How does the FOIL method relate to the distributive property in algebra?

**Tip:** Always double-check your work by re-multiplying the expanded expression to ensure it matches the original expression.

Learn how to expand the expression (x - 2y)(3x + y) using the distributive property. Follow a detailed step-by-step solution with examples. Suitable for students in Grades 7-9 studying algebra and polynomials.

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