Let's solve the expression \(-(2u - 3v)(4v + 3u)\) using the distributive property (also known as FOIL for binomials).

\[
-(2u - 3v)(4v + 3u)
\]

First, distribute each term:

\[
-(2u(4v + 3u) - 3v(4v + 3u))
\]

Now, expand each term:

\[
-(2u \cdot 4v + 2u \cdot 3u - 3v \cdot 4v - 3v \cdot 3u)
\]

This simplifies to:

\[
-(8uv + 6u^2 - 12v^2 - 9vu)
\]

Now, combine like terms:

\[
-(8uv - 9uv + 6u^2 - 12v^2)
\]

Simplifying:

\[
-(-uv + 6u^2 - 12v^2)
\]

Finally, distribute the negative sign:

\[
uv - 6u^2 + 12v^2
\]

So, the simplified expression is:

\[
uv - 6u^2 + 12v^2
\]

### Would you like more details on any steps or have any questions?

Here are 5 related questions to explore:
1. What is the distributive property and how is it applied in algebra?
2. How do you combine like terms when simplifying algebraic expressions?
3. Can you solve \((2u - 3v)^2\)?
4. What are binomials, and how do you multiply them?
5. How do you factor quadratic expressions like \(6u^2 - uv + 12v^2\)?

**Tip:** Always remember to combine like terms carefully when simplifying algebraic expressions!

Learn how to simplify the algebraic expression (2u-3v)(4v+3u) by applying the distributive property and combining like terms. This detailed step-by-step solution covers key concepts from algebra, including binomial multiplication. Suitable for students in Grades 8-10.

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