https://r2.erweima.ai/mathbot/image/17243693937039wf9w3n7.jpg

The expression in the image is:

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

To simplify this expression, we'll follow these steps:

1. **Factor the numerator and the denominator:**

   - **Numerator:** \( x^2 - 8x + 12 \)
     - We need to factor this quadratic expression. We're looking for two numbers that multiply to \( 12 \) (the constant term) and add to \( -8 \) (the coefficient of \( x \)).
     - The numbers \( -6 \) and \( -2 \) satisfy these conditions.
     - Therefore, \( x^2 - 8x + 12 = (x - 6)(x - 2) \).
   
   - **Denominator:** \( x^2 - 36 \)
     - This is a difference of squares, which can be factored as:
     - \( x^2 - 36 = (x - 6)(x + 6) \).

2. **Simplify the expression by canceling out common factors:**

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

   Since \( (x - 6) \) is a common factor in both the numerator and the denominator, it can be canceled out (as long as \( x \neq 6 \)):

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

So, the simplified form of the expression is:

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

**Would you like to go over any of the steps in more detail, or do you have any questions?**

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Here are 5 related questions:

1. What would the expression be if the original denominator was \( x^2 + 36 \) instead of \( x^2 - 36 \)?
2. How would you solve the expression \( \frac{x^2 - 8x + 12}{x^2 + 36} \)?
3. Can you factor \( x^2 - 6x + 9 \), and how would that change the simplification process?
4. What happens if you substitute \( x = 6 \) in the original and simplified expressions?
5. Can you explain the concept of difference of squares and provide another example?

**Tip:** When simplifying rational expressions, always check if the expression can be factored and simplify by canceling out common factors.

Learn how to simplify the rational expression (x^2 - 8x + 12) / (x^2 - 36) step-by-step. This problem involves factoring quadratic expressions and simplifying rational functions by canceling out common factors. Suitable for high school students in Grades 10-12 studying algebra and rational expressions.

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