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!