https://file.aiquickdraw.com/mathbot/file/1727056340799milj187s.jpg

The problem shown in the image asks to simplify the expression:

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

### Step-by-step simplification:

1. **Factor the numerator**: The numerator \( a^2 - 36 \) is a difference of squares, which can be factored as:
   \[
   a^2 - 36 = (a - 6)(a + 6)
   \]

2. **Factor the denominator**: The denominator \( a^2 + 8a + 12 \) is a quadratic expression. We need to factor it:
   - We are looking for two numbers that multiply to \( 12 \) and add to \( 8 \). These numbers are \( 6 \) and \( 2 \).
   \[
   a^2 + 8a + 12 = (a + 6)(a + 2)
   \]

3. **Simplify the fraction**: Now that both the numerator and the denominator are factored, we can cancel out the common factor \( (a + 6) \).
   \[
   \frac{(a - 6)(a + 6)}{(a + 6)(a + 2)} = \frac{a - 6}{a + 2} \quad \text{(as long as \( a \neq -6 \) to avoid division by zero)}
   \]

Thus, the simplified expression is:

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

### Correct answer: **Option A**.

Would you like further details or clarification? Here are some related questions:

1. How can you factor quadratics more efficiently?
2. What happens if \( a = -6 \)? How would the solution change?
3. How do you identify common factors in a rational expression?
4. What are the key properties of the difference of squares?
5. What are some examples of real-world problems using rational expressions?

**Tip**: Always check for restrictions in the variable values when simplifying rational expressions, to ensure no division by zero occurs!

This algebra problem involves simplifying the rational expression (a^2 - 36) / (a^2 + 8a + 12) by factoring both the numerator and the denominator. Key steps include using the difference of squares and factoring quadratics. Suitable for students in Grades 9-10.

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