8𝑥2𝑦(5𝑥2 − 4𝑥𝑦 + 𝑦2 − 3)

Let's expand and simplify the expression \( 8x^2y(5x^2 - 4xy + y^2 - 3) \).

### Step-by-Step Expansion

1. **Distribute \( 8x^2y \) to each term inside the parentheses**:

   \[
   8x^2y \times 5x^2 = 40x^4y
   \]

   \[
   8x^2y \times -4xy = -32x^3y^2
   \]

   \[
   8x^2y \times y^2 = 8x^2y^3
   \]

   \[
   8x^2y \times -3 = -24x^2y
   \]

2. **Combine all the terms**:

   \[
   40x^4y - 32x^3y^2 + 8x^2y^3 - 24x^2y
   \]

### Final Answer:

\[
40x^4y - 32x^3y^2 + 8x^2y^3 - 24x^2y
\]

This is the expanded and simplified form of the given expression.

Would you like further details on any part of the process or have additional questions?

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Here are 5 related questions:

1. How do you factor a polynomial after expanding it?
2. What is the importance of like terms when simplifying expressions?
3. Can you explain the distributive property in algebra?
4. How would this expansion change if the expression inside the parentheses was a sum of squares?
5. What are the potential errors to watch out for when distributing a term over a polynomial?

**Tip:** Always double-check each step of your distribution to ensure that all terms have been correctly multiplied and no sign errors have been made.

Learn how to expand and simplify the expression 8x^2y(5x^2 - 4xy + y^2 - 3) with a detailed step-by-step solution. This problem involves algebraic manipulation of polynomials and the distributive property. Suitable for high school students and above.

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