To determine which rational expression has a restriction of \( x \neq -1 \) in its domain, we need to identify expressions where the denominator equals zero when \( x = -1 \).
Let's analyze each option:
A.
\[
\frac{x^3}{x + 1}
\]
The denominator is \( x + 1 \). Setting this equal to zero:
\[
x + 1 = 0 \implies x = -1
\]
Thus, this expression has a restriction of \( x \neq -1 \).
B.
\[
\frac{2x + 2}{x}
\]
The denominator is \( x \). Setting this equal to zero:
\[
x = 0
\]
This expression has a restriction of \( x \neq 0 \).
C.
\[
\frac{6x + 6}{x^2 + 6x + 5}
\]
The denominator is \( x^2 + 6x + 5 \). Factoring the quadratic:
\[
x^2 + 6x + 5 = (x + 1)(x + 5)
\]
Setting this equal to zero:
\[
(x + 1)(x + 5) = 0 \implies x = -1 \text{ or } x = -5
\]
This expression has a restriction of \( x \neq -1 \) and \( x \neq -5 \).
D.
\[
\frac{x}{6x^3 + 60}
\]
The denominator is \( 6x^3 + 60 \). Setting this equal to zero:
\[
6x^3 + 60 = 0 \implies 6x^3 = -60 \implies x^3 = -10 \implies x = -\sqrt[3]{10}
\]
Thus, there is no restriction at \( x = -1 \).
Conclusion:
Both options A and C have restrictions of \( x \neq -1 \) in their domain. However, the question looks for only one option. Since the explicit mention of the restriction is required and since both A and C fulfill that requirement, the best fit for the answer that represents a simplistic understanding would be option A.
Therefore, the answer is: A. \( \frac{x^3}{x + 1} \).