## Step 1: Define the variables and their corresponding symbolic notation
Let's denote the hours worked by Ringo, Hank, John, and Bill as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is: $8x_0^2 + x_0x_2 + 5x_0x_3 + 4x_1^2 + x_2^2 + 4x_0 + 3x_1 + 5x_2 + 5x_3$.

## Step 3: List the constraints in symbolic notation
Constraints:
1. $5x_0 \leq 143$
2. $7x_0 \leq 140$
3. $5x_1 \leq 143$
4. $5x_1 \leq 140$
5. $x_2 \leq 143$
6. $7x_2 \leq 140$
7. $2x_3 \leq 143$
8. $11x_3 \leq 140$
9. $5x_0^2 + 5x_1^2 \geq 35$
10. $5x_1 + 2x_3 \geq 19$
11. $5x_0^2 + x_2^2 \geq 27$
12. $5x_1 + x_2 \geq 23$
13. $x_2 + 2x_3 \geq 35$
14. $5x_0 + x_2 + 2x_3 \geq 21$
15. $5x_0^2 + 5x_1^2 + x_2^2 \geq 21$
16. $5x_0 + 5x_1 + 2x_3 \geq 21$
17. $5x_1 + x_2 + 2x_3 \geq 21$
18. $5x_0 + x_2 + 2x_3 \geq 30$
19. $5x_0^2 + 5x_1^2 + x_2^2 \geq 30$
20. $5x_0 + 5x_1 + 2x_3 \geq 30$
21. $5x_1^2 + x_2^2 + 4x_3^2 \geq 30$
22. $5x_0 + x_2 + 2x_3 \geq 27$
23. $5x_0^2 + 5x_1^2 + x_2^2 \geq 27$
24. $5x_0 + 5x_1 + 2x_3 \geq 27$
25. $5x_1 + x_2 + 2x_3 \geq 27$
26. $5x_0^2 + x_2^2 + 4x_3^2 \geq 29$
27. $5x_0 + 5x_1 + x_2 \geq 29$
28. $5x_0 + 5x_1 + 2x_3 \geq 29$
29. $5x_1^2 + x_2^2 + 4x_3^2 \geq 29$
30. $5x_1 + 11x_3 \geq 33$
31. $5x_1 + 7x_2 \geq 33$
32. $7x_0 + 11x_3 \geq 25$
33. $5x_0 + 5x_1 \leq 45$
34. $5x_1 + x_2 \leq 94$
35. $5x_1 + 2x_3 \leq 47$
36. $5x_0^2 + x_2^2 \leq 83$
37. $x_2 + 2x_3 \leq 100$
38. $5x_0^2 + 5x_1^2 + 4x_3^2 \leq 66$
39. $5x_0 + 5x_1 + x_2 + 2x_3 \leq 66$
40. $25x_1^2 + 121x_3^2 \leq 92$
41. $7x_0 + 11x_3 \leq 113$
42. $7x_0 + 7x_2 + 11x_3 \leq 93$
43. $7x_0 + 5x_1 + 7x_2 + 11x_3 \leq 93$

## Step 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Ringo'),
        ('x1', 'hours worked by Hank'),
        ('x2', 'hours worked by John'),
        ('x3', 'hours worked by Bill')
    ],
    'objective_function': '8*x0^2 + x0*x2 + 5*x0*x3 + 4*x1^2 + x2^2 + 4*x0 + 3*x1 + 5*x2 + 5*x3',
    'constraints': [
        '5*x0 <= 143',
        '7*x0 <= 140',
        '5*x1 <= 143',
        '5*x1 <= 140',
        'x2 <= 143',
        '7*x2 <= 140',
        '2*x3 <= 143',
        '11*x3 <= 140',
        '5*x0^2 + 5*x1^2 >= 35',
        '5*x1 + 2*x3 >= 19',
        '5*x0^2 + x2^2 >= 27',
        '5*x1 + x2 >= 23',
        'x2 + 2*x3 >= 35',
        '5*x0 + x2 + 2*x3 >= 21',
        '5*x0^2 + 5*x1^2 + x2^2 >= 21',
        '5*x0 + 5*x1 + 2*x3 >= 21',
        '5*x1 + x2 + 2*x3 >= 21',
        '5*x0 + x2 + 2*x3 >= 30',
        '5*x0^2 + 5*x1^2 + x2^2 >= 30',
        '5*x0 + 5*x1 + 2*x3 >= 30',
        '5*x1^2 + x2^2 + 4*x3^2 >= 30',
        '5*x0 + x2 + 2*x3 >= 27',
        '5*x0^2 + 5*x1^2 + x2^2 >= 27',
        '5*x0 + 5*x1 + 2*x3 >= 27',
        '5*x1 + x2 + 2*x3 >= 27',
        '5*x0^2 + x2^2 + 4*x3^2 >= 29',
        '5*x0 + 5*x1 + x2 >= 29',
        '5*x0 + 5*x1 + 2*x3 >= 29',
        '5*x1^2 + x2^2 + 4*x3^2 >= 29',
        '5*x1 + 11*x3 >= 33',
        '5*x1 + 7*x2 >= 33',
        '7*x0 + 11*x3 >= 25',
        '5*x0 + 5*x1 <= 45',
        '5*x1 + x2 <= 94',
        '5*x1 + 2*x3 <= 47',
        '5*x0^2 + x2^2 <= 83',
        'x2 + 2*x3 <= 100',
        '5*x0^2 + 5*x1^2 + 4*x3^2 <= 66',
        '5*x0 + 5*x1 + x2 + 2*x3 <= 66',
        '25*x1^2 + 121*x3^2 <= 92',
        '7*x0 + 11*x3 <= 113',
        '7*x0 + 7*x2 + 11*x3 <= 93',
        '7*x0 + 5*x1 + 7*x2 + 11*x3 <= 93'
    ]
}
```

## Step 5: Write the Gurobi code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name='x0', vtype=gurobi.GRB.INTEGER)  # hours worked by Ringo
    x1 = model.addVar(name='x1')  # hours worked by Hank
    x2 = model.addVar(name='x2', vtype=gurobi.GRB.INTEGER)  # hours worked by John
    x3 = model.addVar(name='x3')  # hours worked by Bill

    # Objective function
    model.setObjective(8 * x0**2 + x0 * x2 + 5 * x0 * x3 + 4 * x1**2 + x2**2 + 4 * x0 + 3 * x1 + 5 * x2 + 5 * x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(5 * x0 <= 143)
    model.addConstr(7 * x0 <= 140)
    model.addConstr(5 * x1 <= 143)
    model.addConstr(5 * x1 <= 140)
    model.addConstr(x2 <= 143)
    model.addConstr(7 * x2 <= 140)
    model.addConstr(2 * x3 <= 143)
    model.addConstr(11 * x3 <= 140)
    model.addConstr(5 * x0**2 + 5 * x1**2 >= 35)
    model.addConstr(5 * x1 + 2 * x3 >= 19)
    model.addConstr(5 * x0**2 + x2**2 >= 27)
    model.addConstr(5 * x1 + x2 >= 23)
    model.addConstr(x2 + 2 * x3 >= 35)
    model.addConstr(5 * x0 + x2 + 2 * x3 >= 21)
    model.addConstr(5 * x0**2 + 5 * x1**2 + x2**2 >= 21)
    model.addConstr(5 * x0 + 5 * x1 + 2 * x3 >= 21)
    model.addConstr(5 * x1 + x2 + 2 * x3 >= 21)
    model.addConstr(5 * x0 + x2 + 2 * x3 >= 30)
    model.addConstr(5 * x0**2 + 5 * x1**2 + x2**2 >= 30)
    model.addConstr(5 * x0 + 5 * x1 + 2 * x3 >= 30)
    model.addConstr(5 * x1**2 + x2**2 + 4 * x3**2 >= 30)
    model.addConstr(5 * x0 + x2 + 2 * x3 >= 27)
    model.addConstr(5 * x0**2 + 5 * x1**2 + x2**2 >= 27)
    model.addConstr(5 * x0 + 5 * x1 + 2 * x3 >= 27)
    model.addConstr(5 * x1 + x2 + 2 * x3 >= 27)
    model.addConstr(5 * x0**2 + x2**2 + 4 * x3**2 >= 29)
    model.addConstr(5 * x0 + 5 * x1 + x2 >= 29)
    model.addConstr(5 * x0 + 5 * x1 + 2 * x3 >= 29)
    model.addConstr(5 * x1**2 + x2**2 + 4 * x3**2 >= 29)
    model.addConstr(5 * x1 + 11 * x3 >= 33)
    model.addConstr(5 * x1 + 7 * x2 >= 33)
    model.addConstr(7 * x0 + 11 * x3 >= 25)
    model.addConstr(5 * x0 + 5 * x1 <= 45)
    model.addConstr(5 * x1 + x2 <= 94)
    model.addConstr(5 * x1 + 2 * x3 <= 47)
    model.addConstr(5 * x0**2 + x2**2 <= 83)
    model.addConstr(x2 + 2 * x3 <= 100)
    model.addConstr(5 * x0**2 + 5 * x1**2 + 4 * x3**2 <= 66)
    model.addConstr(5 * x0 + 5 * x1 + x2 + 2 * x3 <= 66)
    model.addConstr(25 * x1**2 + 121 * x3**2 <= 92)
    model.addConstr(7 * x0 + 11 * x3 <= 113)
    model.addConstr(7 * x0 + 7 * x2 + 11 * x3 <= 93)
    model.addConstr(7 * x0 + 5 * x1 + 7 * x2 + 11 * x3 <= 93)

    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Objective: ", model.objVal)
        print("x0: ", x0.varValue)
        print("x1: ", x1.varValue)
        print("x2: ", x2.varValue)
        print("x3: ", x3.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```