Let's simplify the expression \( (3x^4y - 2xy^2) \times (-1.5x^2y^3) \). 

### Step-by-Step Simplification:

This expression involves distributing the multiplication over the terms inside the parentheses. Let's break it down:

\[
(3x^4y - 2xy^2) \times (-1.5x^2y^3)
\]

1. **Distribute** the multiplication:

   \[
   = (3x^4y) \times (-1.5x^2y^3) + (-2xy^2) \times (-1.5x^2y^3)
   \]

2. **Multiply the first term** \( 3x^4y \times -1.5x^2y^3 \):
   - Coefficient: \( 3 \times -1.5 = -4.5 \)
   - \( x^4 \times x^2 = x^{4+2} = x^6 \)
   - \( y \times y^3 = y^{1+3} = y^4 \)
   
   So, the first term becomes:
   \[
   -4.5x^6y^4
   \]

3. **Multiply the second term** \( -2xy^2 \times -1.5x^2y^3 \):
   - Coefficient: \( -2 \times -1.5 = 3 \)
   - \( x \times x^2 = x^{1+2} = x^3 \)
   - \( y^2 \times y^3 = y^{2+3} = y^5 \)
   
   So, the second term becomes:
   \[
   3x^3y^5
   \]

4. **Combine the results**:

\[
-4.5x^6y^4 + 3x^3y^5
\]

### Final Answer:

\[
-4.5x^6y^4 + 3x^3y^5
\]

Would you like any further explanation on the steps? Here are five related questions to expand your understanding:

1. How would the result change if the coefficient of the second term was positive \( +1.5x^2y^3 \) instead of negative?
2. What happens if the expression \( x^2y^3 \) is raised to a different power before multiplying?
3. How would the expression change if you added an additional term inside the first parentheses, like \( +5xy^3 \)?
4. How would you factor the resulting expression \( -4.5x^6y^4 + 3x^3y^5 \) if possible?
5. Can you multiply the original expression by another binomial, such as \( (x^2 - y^2) \)?

**Tip:** When multiplying terms with exponents, remember to add the exponents of the like bases!

Learn how to simplify the expression (3x^4y - 2xy^2) * (-1.5x^2y^3) with a detailed step-by-step solution. This problem involves multiplication of polynomials and exponents. Suitable for students in Grades 9-12.

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