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To simplify the given expression:

\[
\frac{x^2 - 8x + 12}{x^2 - 36}
\]

### Step 1: Factor the numerator
The numerator is \( x^2 - 8x + 12 \).

- To factor this quadratic expression, look for two numbers that multiply to \( 12 \) and add to \( -8 \). These numbers are \( -6 \) and \( -2 \).
- Therefore, the numerator factors as:
  \[
  (x - 6)(x - 2)
  \]

### Step 2: Factor the denominator
The denominator is \( x^2 - 36 \).

- Notice that \( x^2 - 36 \) is a difference of squares, which can be factored as:
  \[
  (x - 6)(x + 6)
  \]

### Step 3: Simplify the expression
Now the expression becomes:
\[
\frac{(x - 6)(x - 2)}{(x - 6)(x + 6)}
\]

- The \( (x - 6) \) terms in the numerator and denominator cancel out, leaving:
  \[
  \frac{x - 2}{x + 6}
  \]

So, the simplified expression is:

\[
\frac{x - 2}{x + 6}
\]

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Would you like further explanation on this process, or do you have any additional questions?

Here are some related questions to consider:
1. What happens if \( x = 6 \) in the original expression?
2. How can you verify the simplification by expanding the simplified expression?
3. What if the original quadratic expressions couldn't be factored easily?
4. How would this expression change if there were additional terms in the numerator or denominator?
5. What is the domain of the simplified expression?

**Tip:** When simplifying rational expressions, always check for common factors in the numerator and denominator. After factoring, ensure to cancel out any common terms.

Learn how to simplify the expression (x^2 - 8x + 12) / (x^2 - 36) step-by-step. This problem involves factoring quadratic expressions and applying the difference of squares theorem. Suitable for students in Grades 7-9.

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