To expand the expression \((3x - 4y)(2x + 5y)\), you can use the distributive property (also known as the FOIL method for binomials):

\[
(a + b)(c + d) = ac + ad + bc + bd
\]

In your case:
- \(a = 3x\)
- \(b = -4y\)
- \(c = 2x\)
- \(d = 5y\)

Now, apply the distributive property:

\[
(3x - 4y)(2x + 5y) = 3x \cdot 2x + 3x \cdot 5y - 4y \cdot 2x - 4y \cdot 5y
\]

Simplify each term:

\[
3x \cdot 2x = 6x^2
\]

\[
3x \cdot 5y = 15xy
\]

\[
-4y \cdot 2x = -8xy
\]

\[
-4y \cdot 5y = -20y^2
\]

Now, combine like terms:

\[
6x^2 + 15xy - 8xy - 20y^2
\]

\[
6x^2 + 7xy - 20y^2
\]

So, the expanded form is:

\[
(3x - 4y)(2x + 5y) = 6x^2 + 7xy - 20y^2
\]

Would you like any further details or have any questions about this solution?

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Here are five related questions:

1. How would you expand \((x + 2)(x - 3)\)?
2. What is the result of expanding \((2a - b)(3a + 4b)\)?
3. How can you factor the expression \(6x^2 + 7xy - 20y^2\) back into two binomials?
4. What happens if both binomials have the same signs, e.g., \((3x + 4y)(2x + 5y)\)?
5. Can you expand \((3x - 4y)^2\) and compare it to the original expression?

**Tip:** When using the FOIL method, always combine like terms to simplify your final expression!

Learn how to expand the expression (3x - 4y)(2x + 5y) using the distributive property (FOIL method). This problem covers algebraic concepts such as polynomials and the distributive property, providing a step-by-step solution. Suitable for students in Grades 7-9.

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