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To solve the problem, we need to determine which of the given rational expressions has a domain restriction where \( x \neq -1 \). A rational expression's domain is restricted where the denominator equals zero because division by zero is undefined.

### Let's analyze each option:

1. **Option 1:** \(\frac{x^2 + 6x + 5}{6x + 6}\)
   - The denominator is \(6x + 6\). Setting it equal to zero:
     \[
     6x + 6 = 0 \implies x = -1
     \]
   - The domain restriction is \(x \neq -1\), which matches the condition.

2. **Option 2:** \(\frac{\sqrt{x}}{2x + 2}\)
   - The denominator is \(2x + 2\). Setting it equal to zero:
     \[
     2x + 2 = 0 \implies x = -1
     \]
   - This also results in \(x \neq -1\), but note that the numerator \(\sqrt{x}\) also imposes a restriction \(x \geq 0\), so \(x \neq -1\) might not be the only restriction. 

3. **Option 3:** \(\frac{x^2 - x - 6}{x}\)
   - The denominator is \(x\). Setting it equal to zero:
     \[
     x = 0
     \]
   - The domain restriction is \(x \neq 0\), not \(x \neq -1\).

4. **Option 4:** \(\frac{x - 1}{x^2}\)
   - The denominator is \(x^2\). Setting it equal to zero:
     \[
     x^2 = 0 \implies x = 0
     \]
   - The domain restriction is \(x \neq 0\), not \(x \neq -1\).

### Conclusion:
The correct answer is **Option 1: \(\frac{x^2 + 6x + 5}{6x + 6}\)**, since the domain restriction is \(x \neq -1\).

Would you like any further details on this problem, or do you have any other questions?

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Here are five related questions you might find interesting:
1. How do you determine the domain of a rational function in general?
2. What are the steps to simplify a rational expression?
3. How do you find the roots of a quadratic expression?
4. What is the difference between a rational expression and a polynomial?
5. How can square roots affect the domain of a function?

**Tip:** Always factor both the numerator and denominator when analyzing the domain of a rational expression, as this can help identify any common factors that might simplify the expression and change the domain restrictions.

Which of the following is a rational expression whose domain has a restriction of x ≠ -1?

Learn how to identify domain restrictions in rational expressions with a focus on x ≠ -1. This algebra problem is suitable for students in Grades 9-11 and covers key concepts such as rational expressions, domain restrictions, and the implications of division by zero.

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