Find all numbers $x \in \mathbb Z$ for which the number
\[x^4 + x^3 + x^2 + x + 1\]
is a perfect square.
To find all integers \( x \in \mathbb{Z} \) for which the expression \( x^4 + x^3 + x^2 + x + 1 \) is a perfect square, we will analyze the given polynomial and check specific cases.

1. **Assume \( y^2 = x^4 + x^3 + x^2 + x + 1 \)**:
   We need to determine when this expression is a perfect square.

2. **Case 1: \( x \) is even and \( x \neq 0 \)**:
   Let \( x = 2k \) for some integer \( k \). Then:
   \[
   x^4 + x^3 + x^2 + x + 1 = (2k)^4 + (2k)^3 + (2k)^2 + 2k + 1
   \]
   We need to check if this can be a perfect square. Consider the bounds:
   \[
   (x^2 + \frac{x}{2} - 1)^2 < y^2 < (x^2 + \frac{x}{2} + 1)^2
   \]
   Simplifying the bounds:
   \[
   (4k^2 + k - 1)^2 < y^2 < (4k^2 + k + 1)^2
   \]
   Since \( y^2 \) must be an integer, it must be exactly one of these squares, but it cannot be because there is no integer \( y \) such that \( y^2 \) lies strictly between two consecutive squares. Thus, no solution exists for even \( x \neq 0 \).

3. **Case 2: \( x \) is odd and \( x \notin \{-1, 1, 3\} \)**:
   Let \( x = 2k + 1 \) for some integer \( k \). Then:
   \[
   x^4 + x^3 + x^2 + x + 1 = (2k+1)^4 + (2k+1)^3 + (2k+1)^2 + (2k+1) + 1
   \]
   We need to check if this can be a perfect square. Consider the bounds:
   \[
   (x^2 + \frac{x-1}{2})^2 < y^2 < (x^2 + \frac{x+1}{2})^2
   \]
   Simplifying the bounds:
   \[
   ((2k+1)^2 + k)^2 < y^2 < ((2k+1)^2 + k + 1)^2
   \]
   Again, since \( y^2 \) must be an integer, it must be exactly one of these squares, but it cannot be because there is no integer \( y \) such that \( y^2 \) lies strictly between two consecutive squares. Thus, no solution exists for odd \( x \notin \{-1, 1, 3\} \).

4. **Check specific values \( x = -1, 0, 1, 3 \)**:
   - For \( x = -1 \):
     \[
     (-1)^4 + (-1)^3 + (-1)^2 + (-1) + 1 = 1 - 1 + 1 - 1 + 1 = 1
     \]
     \( 1 \) is a perfect square.
   - For \( x = 0 \):
     \[
     0^4 + 0^3 + 0^2 + 0 + 1 = 1
     \]
     \( 1 \) is a perfect square.
   - For \( x = 1 \):
     \[
     1^4 + 1^3 + 1^2 + 1 + 1 = 1 + 1 + 1 + 1 + 1 = 5
     \]
     \( 5 \) is not a perfect square.
   - For \( x = 3 \):
     \[
     3^4 + 3^3 + 3^2 + 3 + 1 = 81 + 27 + 9 + 3 + 1 = 121
     \]
     \( 121 = 11^2 \) is a perfect square.

Thus, the only values of \( x \) for which \( x^4 + x^3 + x^2 + x + 1 \) is a perfect square are \( x = -1, 0, 3 \).

The final answer is \( \boxed{-1, 0, 3} \).