Sure! Let's create a test case generator for the given problem statement where you need to compute \(a + a^2 + a^3\) for a given integer \(a\).

Here's the Python code for the test case generator:

```python
import random

def tcgen():
    # Generate a random integer a within the given constraints
    a = random.randint(1, 10)
    
    return a

# Example usage:
test_case = tcgen()
print(f"Input: {test_case}")
# For verification purposes, compute the expected output
expected_output = test_case + test_case**2 + test_case**3
print(f"Output: {expected_output}")
```

This generator will produce a random integer \(a\) in the range \(1 \leq a \leq 10\), as specified in the problem constraints. The printed output includes the test input and the expected result based on the formula \(a + a^2 + a^3\). 

You can run this code to generate multiple test cases and verify that the output matches the expected results.