To determine which of the given options has a domain restriction of \( x \neq -1 \), we need to look for an expression that has a denominator that becomes zero when \( x = -1 \).
Let's analyze each option:
-
\( \frac{x^2 + 6x + 5}{6x + 6} \):
- Denominator: \( 6x + 6 = 6(x + 1) \)
- This is zero when \( x = -1 \), so there is a restriction \( x \neq -1 \).
-
\( \frac{\sqrt{x}}{2x + 2} \):
- Denominator: \( 2x + 2 = 2(x + 1) \)
- This is also zero when \( x = -1 \), so there is a restriction \( x \neq -1 \).
-
\( \frac{x + 1}{x^3} \):
- Denominator: \( x^3 \)
- This is zero when \( x = 0 \), so there is no restriction for \( x \neq -1 \).
-
\( 6x^3 + 60x \):
- This is an algebraic expression, not a rational expression (no denominator), so there are no restrictions.
Since both the first and second expressions produce the restriction \( x \neq -1 \), the first two options are both valid answers. However, if you are looking for only one option, the first option is typically preferred.
Therefore, the correct answer is:
\( \frac{x^2 + 6x + 5}{6x + 6} \) (with the restriction \( x \neq -1 \)).