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Let's solve the given expression: 

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

### Step 1: Expand both squares
For \((a - 3b)^2\), use the expansion formula \((x - y)^2 = x^2 - 2xy + y^2\):

\[
(a - 3b)^2 = a^2 - 2a(3b) + (3b)^2 = a^2 - 6ab + 9b^2
\]

For \((a + b)^2\), use the expansion formula \((x + y)^2 = x^2 + 2xy + y^2\):

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

### Step 2: Subtract the two expanded forms
Now, subtract \((a + b)^2\) from \((a - 3b)^2\):

\[
(a - 3b)^2 - (a + b)^2 = \left( a^2 - 6ab + 9b^2 \right) - \left( a^2 + 2ab + b^2 \right)
\]

Distribute the negative sign:

\[
= a^2 - 6ab + 9b^2 - a^2 - 2ab - b^2
\]

### Step 3: Combine like terms
Cancel out \(a^2\) and combine the remaining terms:

\[
= (-6ab - 2ab) + (9b^2 - b^2)
\]

\[
= -8ab + 8b^2
\]

### Step 4: Final Answer
Thus, the simplified expression is:

\[
8b^2 - 8ab
\]

This matches option **A** in the provided choices.

Would you like further details on any step?

Here are some related questions to practice:
1. How do you expand binomials using the square of a sum or difference formula?
2. What is the general formula for \((x - y)^2\) and \((x + y)^2\)?
3. How do you simplify expressions with like terms?
4. Can you expand and simplify \((2x - 5y)^2 - (x + 4y)^2\)?
5. How does changing the signs of terms in a binomial affect the result?

**Tip:** When simplifying expressions involving squares of binomials, always expand each term separately before combining like terms.

Learn how to simplify the expression (a - 3b)^2 - (a + b)^2 using binomial expansion. This detailed solution explains key algebraic concepts, such as expanding binomials and combining like terms. Suitable for students in Grades 8-10.

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