Let's simplify the given expression:

\[
7a(a - 3ab) + a^2 + 8a^2b
\]

### Step 1: Distribute the terms in the first part

Distribute \(7a\) over \((a - 3ab)\):

\[
7a(a - 3ab) = 7a^2 - 21a^2b
\]

### Step 2: Combine all the terms

Now rewrite the entire expression with the distributed terms:

\[
7a^2 - 21a^2b + a^2 + 8a^2b
\]

### Step 3: Combine like terms

Group the \(a^2\) terms and the \(a^2b\) terms:

\[
(7a^2 + a^2) + (-21a^2b + 8a^2b)
\]

Simplify both groups:

\[
8a^2 - 13a^2b
\]

### Final Simplified Expression:

\[
8a^2 - 13a^2b
\]

This is the simplified form of the given expression.

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you factor common terms from algebraic expressions?
2. What is the difference between distributing and factoring in algebra?
3. How would you handle expressions with higher powers of \(a\)?
4. Can you factor the final expression \(8a^2 - 13a^2b\)?
5. How would the expression change if there were negative exponents involved?

**Tip**: Always group similar terms when simplifying algebraic expressions to avoid missing simplifications!

Learn how to simplify the algebraic expression 7a(a - 3ab) + a² + 8a²b step by step. This problem covers key algebraic concepts, including the distributive property and combining like terms. Ideal for students in Grades 8-10.

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