Let's expand the expression \( 2x^2y(x^3y + \frac{4}{xy}) \).

### Step-by-Step Solution:

1. **Distribute** \( 2x^2y \) to each term inside the parentheses:
   \[
   2x^2y \cdot x^3y + 2x^2y \cdot \frac{4}{xy}
   \]

2. **First term**: Multiply \( 2x^2y \) by \( x^3y \):
   \[
   2x^2y \cdot x^3y = 2x^{2+3}y^{1+1} = 2x^5y^2
   \]

3. **Second term**: Multiply \( 2x^2y \) by \( \frac{4}{xy} \):
   \[
   2x^2y \cdot \frac{4}{xy} = \frac{2x^2y \cdot 4}{xy} = 2 \cdot \frac{4x^2y}{xy}
   \]
   Simplify the powers of \( x \) and \( y \):
   \[
   \frac{4x^2}{x} = 4x \quad \text{and} \quad \frac{y}{y} = 1
   \]
   So the second term simplifies to:
   \[
   2 \cdot 4x = 8x
   \]

4. **Final expression**: Combine the two results:
   \[
   2x^5y^2 + 8x
   \]

Thus, the expanded form of the expression is:
\[
2x^5y^2 + 8x
\]

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### Related Questions:
1. How do you expand binomials with higher powers?
2. Can you explain how to handle distribution with negative terms?
3. What happens if one of the variables is a fraction in the expression?
4. How do you factor an expression like \( 2x^5y^2 + 8x \)?
5. How do you simplify expressions with fractional exponents?

### Tip:
Always check if terms can be further simplified after expansion, especially when variables cancel out.

Learn how to expand and simplify the expression 2x^2y(x^3y + 4/xy) using the distributive property and laws of exponents. This problem involves basic algebra and polynomials, suitable for students in Grades 9-11.

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