Let's break down the expression $15uv^2(3u - v) + 9u^2v(v - 3u)$ step by step.

Expression:

$15uv^2(3u - v) + 9u^2v(v - 3u)$

Step 1: Expand both terms individually.

First term: $15uv^2(3u - v)$

Distribute $15uv^2$ to each term inside the parentheses:

$15uv^2(3u) - 15uv^2(v)$

This becomes:

$45u^2v^2 - 15uv^3$

Second term: $9u^2v(v - 3u)$

Similarly, distribute $9u^2v$ to each term inside the parentheses:

$9u^2v(v) - 9u^2v(3u)$

This simplifies to:

$9u^2v^2 - 27u^3v$

Step 2: Combine both expanded results.

Now, combine the two expanded expressions:

$(45u^2v^2 - 15uv^3) + (9u^2v^2 - 27u^3v)$

Step 3: Group like terms.

Group the terms with similar powers of $u$ and $v$:

• $45u^2v^2 + 9u^2v^2 = 54u^2v^2$
• The term $-15uv^3$ stands alone.
• The term $-27u^3v$ stands alone.

So the expression becomes:

$54u^2v^2 - 15uv^3 - 27u^3v$

Final Expression:

The simplified form is:

$54u^2v^2 - 15uv^3 - 27u^3v$

This is as simplified as the expression can get. If needed, you can factor further, but it would not reduce the complexity any further.

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Would you like a deeper explanation or help with another topic?

Relative Questions:

1. How do you factor quadratic expressions involving two variables?
2. What is the importance of grouping like terms in algebraic expressions?
3. Can you explain how to recognize common factors in expressions?
4. How do you simplify polynomial expressions with multiple variables?
5. What are some strategies to check if an expression can be simplified further?

Tip: Always check if there is a common factor among all terms before expanding or simplifying an expression.