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To solve this problem, we need to determine the value of \( f(-2) \) given that the function \( f(x) = \frac{x^2 - 4}{x + 2} \) is continuous for all real numbers.

### Step-by-Step Solution

1. **Simplify the Expression for \( f(x) \):**

   The given function is:
   \[
   f(x) = \frac{x^2 - 4}{x + 2}
   \]

   Notice that the expression is undefined at \( x = -2 \) because the denominator becomes zero. To simplify the expression, factor the numerator:

   \[
   x^2 - 4 = (x - 2)(x + 2)
   \]

   Thus, we can rewrite \( f(x) \) as:

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

   For \( x \neq -2 \), the \( x + 2 \) terms cancel out:

   \[
   f(x) = x - 2 \quad \text{(for } x \neq -2\text{)}
   \]

2. **Find the Limit as \( x \to -2 \):**

   Since \( f(x) \) must be continuous for all real numbers, including at \( x = -2 \), the limit of \( f(x) \) as \( x \to -2 \) must equal \( f(-2) \). Calculate the limit:

   \[
   \lim_{x \to -2} f(x) = \lim_{x \to -2} (x - 2)
   \]

   Substitute \( x = -2 \):

   \[
   \lim_{x \to -2} (x - 2) = -2 - 2 = -4
   \]

3. **Determine the Value of \( f(-2) \):**

   For the function to be continuous at \( x = -2 \), we must have:

   \[
   f(-2) = \lim_{x \to -2} f(x) = -4
   \]

### Conclusion

The value of \( f(-2) \) is \( -4 \). Thus, the correct answer is:

\[
\boxed{\text{A) } -4}
\]

Would you like further details, or do you have any questions?

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Here are 5 related questions for further exploration:
1. How would you define continuity of a function at a point?
2. What happens if a function is not continuous at a point?
3. How do you find the limit of a rational function as it approaches a point of discontinuity?
4. Can a function be continuous but not differentiable at some point?
5. What is the significance of factoring the numerator and denominator in rational functions?

**Tip:** Always check for continuity and differentiability separately; continuity does not always imply differentiability!

If the function f is continuous for all real numbers and if f(x) = (x^2 - 4)/(x + 2) when x ≠ -2, then f(-2) = ?

This problem explores the continuity of the function f(x) = (x^2 - 4)/(x + 2) for x = -2. The solution involves factoring the quadratic and applying the limit to determine the value of f(-2). Suitable for high school students in Grades 11-12.

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