To simplify the expression \((a - 2b)(a^2 + 3ab - 7b^2)\), we'll use the distributive property (also known as the FOIL method for binomials). This means multiplying each term in the first parentheses by each term in the second parentheses:

### Step-by-step simplification:

\[
(a - 2b)(a^2 + 3ab - 7b^2)
\]

1. **Multiply \(a\) by each term inside the parentheses:**
   - \(a \times a^2 = a^3\)
   - \(a \times 3ab = 3a^2b\)
   - \(a \times (-7b^2) = -7ab^2\)

2. **Multiply \(-2b\) by each term inside the parentheses:**
   - \(-2b \times a^2 = -2a^2b\)
   - \(-2b \times 3ab = -6ab^2\)
   - \(-2b \times (-7b^2) = 14b^3\)

Now, putting everything together:

\[
a^3 + 3a^2b - 7ab^2 - 2a^2b - 6ab^2 + 14b^3
\]

### Combine like terms:

- \(3a^2b - 2a^2b = a^2b\)
- \(-7ab^2 - 6ab^2 = -13ab^2\)

Thus, the simplified expression is:

\[
a^3 + a^2b - 13ab^2 + 14b^3
\]

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Would you like any further explanations or additional steps?

Here are five related questions to help you explore the topic further:

1. How would you expand a product of two binomials with different powers of \(a\) and \(b\)?
2. What is the significance of combining like terms in polynomial expressions?
3. How does the distributive property apply to other types of algebraic expressions (e.g., with fractions)?
4. Can you simplify the expression \((x + 3)(x^2 + 5x - 4)\)?
5. What is the general approach to multiplying binomials with both variables and constants?

**Tip:** When simplifying algebraic expressions, always look for like terms that can be combined for a cleaner result!

Learn how to simplify the expression (a − 2b)(a^2 + 3ab − 7b^2) using the distributive property (FOIL method). This detailed solution covers expanding binomials, combining like terms, and understanding algebraic simplifications. Suitable for students in Grades 8-10.

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