To simplify the expression \( 15uv^2(3u - v) + 9u^2v(v - 3u) \), let's follow these steps:

### Step 1: Expand both terms.
- First term: \( 15uv^2(3u - v) \)
  \[
  15uv^2(3u) = 45u^2v^2
  \]
  \[
  15uv^2(-v) = -15uv^3
  \]
  So, the first term expands to:
  \[
  45u^2v^2 - 15uv^3
  \]

- Second term: \( 9u^2v(v - 3u) \)
  \[
  9u^2v(v) = 9u^2v^2
  \]
  \[
  9u^2v(-3u) = -27u^3v
  \]
  So, the second term expands to:
  \[
  9u^2v^2 - 27u^3v
  \]

### Step 2: Combine like terms.
Now we combine the results from both expansions:
\[
(45u^2v^2 - 15uv^3) + (9u^2v^2 - 27u^3v)
\]

Group similar terms:
\[
(45u^2v^2 + 9u^2v^2) - 15uv^3 - 27u^3v
\]
Simplify the coefficients:
\[
54u^2v^2 - 15uv^3 - 27u^3v
\]

### Final simplified expression:
\[
54u^2v^2 - 15uv^3 - 27u^3v
\]

This is the simplified form of the given expression.

Would you like more details on any of the steps? Here are some related questions to expand on:

1. How do you factor expressions like this?
2. What is the role of distributive property in this expansion?
3. Can this expression be factored further?
4. How would you approach similar problems with different powers of variables?
5. How can you identify like terms in algebraic expressions?

**Tip**: Always combine like terms carefully by grouping terms that have the same variables raised to the same powers.

Simplify the algebraic expression 15uv^2(3u - v) + 9u^2v(v - 3u) using distributive property and combining like terms. This detailed step-by-step solution explains polynomial expansion and the process of simplifying complex algebraic terms. Ideal for students in Grades 9-11.

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