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The given expression is:

\[
\frac{(6a - 6b)^2}{6a^2 - 6b^2}
\]

### Step 1: Simplify the numerator
The numerator is \((6a - 6b)^2\). Expanding this expression:

\[
(6a - 6b)^2 = (6a)^2 - 2(6a)(6b) + (6b)^2
\]
\[
= 36a^2 - 72ab + 36b^2
\]

### Step 2: Simplify the denominator
The denominator is \(6a^2 - 6b^2\), which is a difference of squares. It can be factored as:

\[
6a^2 - 6b^2 = 6(a^2 - b^2)
\]
\[
a^2 - b^2 = (a - b)(a + b)
\]
Thus, the denominator becomes:
\[
6(a - b)(a + b)
\]

### Step 3: Rewrite the expression
Now, the expression becomes:

\[
\frac{36a^2 - 72ab + 36b^2}{6(a - b)(a + b)}
\]

### Step 4: Factor the numerator
We can factor out 36 from the numerator:

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

Notice that \(a^2 - 2ab + b^2\) is a perfect square trinomial, so we can factor it as:

\[
36(a - b)^2
\]

### Step 5: Final simplification
Now the expression is:

\[
\frac{36(a - b)^2}{6(a - b)(a + b)}
\]

We can cancel out one \((a - b)\) term from both the numerator and denominator:

\[
\frac{36(a - b)}{6(a + b)}
\]

Finally, divide 36 by 6:

\[
\frac{6(a - b)}{a + b}
\]

### Final Answer:
The simplified expression is:

\[
\frac{6(a - b)}{a + b}
\]

Would you like a more detailed explanation on any step, or have further questions?

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Here are five related questions to explore further:
1. How do you recognize a difference of squares in an expression?
2. What are perfect square trinomials and how do you factor them?
3. Can you explain why we can cancel out terms from the numerator and denominator?
4. What are some common factoring techniques used in simplifying expressions?
5. How would this simplification change if the signs of terms were reversed?

**Tip:** Always check for common factors or patterns like difference of squares or perfect square trinomials when simplifying algebraic expressions.

Learn how to simplify the expression (6a - 6b)^2 / (6a^2 - 6b^2) by factoring the difference of squares and perfect square trinomials. Follow the detailed step-by-step solution, suitable for students in Grades 9-12.

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