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To solve this problem, we need to determine the domain of the function \( f(x) = \frac{x^2 + 2x}{\sqrt{-2x - 6}} \).

### Step 1: Analyze the Square Root Expression
The square root in the denominator, \(\sqrt{-2x - 6}\), must be defined. This requires that the expression inside the square root must be non-negative:

\[
-2x - 6 \geq 0
\]

### Step 2: Solve the Inequality
To find the values of \(x\) that satisfy this inequality, we solve it as follows:

\[
-2x \geq 6
\]
\[
x \leq -3
\]

### Step 3: Consider the Denominator
Since the square root is in the denominator, it cannot be zero. Therefore, we also need to ensure:

\[
\sqrt{-2x - 6} \neq 0
\]

This occurs when \( -2x - 6 = 0 \), which simplifies to:

\[
x = -3
\]

Since \(x = -3\) would make the denominator zero, \(x = -3\) must be excluded from the domain.

### Final Domain
The domain of \(f(x)\) consists of all \(x\) such that:

\[
x < -3
\]

### Conclusion
The correct answer is:

\[
\boxed{A}
\]

This corresponds to the domain \( \{x \in \mathbb{R}, x < -3\} \).

---

Do you have any questions, or would you like further details? Here are some related questions:

1. How do you solve inequalities involving square roots?
2. What happens to a function if the denominator is zero?
3. Can a square root expression in the denominator ever be negative?
4. What is the difference between domain and range in a function?
5. How does excluding certain values from the domain affect the function?

**Tip:** Always check both the numerator and denominator when determining the domain of a function.

Jika f(x) = (x^2 + 2x) / sqrt(-2x - 6), maka daerah asalnya fungsi f adalah?

Learn how to determine the domain of the function f(x) = (x^2 + 2x) / sqrt(-2x - 6). This problem involves understanding square root expressions and solving inequalities, suitable for Grades 10-12.

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