## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by John', 'hours worked by Dale', 'hours worked by Mary', 'hours worked by Bobby'] which can be represented symbolically as $x_0, x_1, x_2, x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $4x_0^2 + 9x_0x_1 + 8x_0x_2 + 9x_0x_3 + 2x_1^2 + 3x_1x_2 + 5x_1x_3 + 4x_2^2 + x_2x_3 + 6x_3^2 + 5x_0 + 6x_1 + x_2 + 9x_3$.

## Step 3: List the constraints in symbolic notation
Constraints:
1. $2x_0 \leq 78$
2. $12x_0 \leq 110$
3. $14x_0 \leq 99$
4. $1x_1 \leq 78$
5. $6x_1 \leq 110$
6. $6x_1 \leq 99$
7. $9x_2 \leq 78$
8. $5x_2 \leq 110$
9. $11x_2 \leq 99$
10. $12x_3 \leq 78$
11. $13x_3 \leq 110$
12. $11x_3 \leq 99$
13. $2x_0^2 + x_1^2 + x_2^2 \geq 10$
14. $2x_0 + x_1 + x_3 \geq 10$
15. $x_1 + x_2 + x_3 \geq 10$
16. $2x_0 + x_1 + x_2 \geq 12$
17. $2x_0 + x_1 + x_3 \geq 12$
18. $x_1^2 + x_2^2 + x_3^2 \geq 12$
19. $2x_0^2 + x_1^2 + x_2^2 \geq 11$
20. $2x_0^2 + x_1^2 + x_3^2 \geq 11$
21. $x_1 + x_2 + x_3 \geq 11$
22. $12x_0 + 6x_1 \geq 14$
23. $12x_0 + 5x_2 \geq 15$
24. $5x_2 + 13x_3 \geq 17$
25. $12x_0 + 13x_3 \geq 15$
26. $14x_0^2 + 6x_1^2 \geq 11$
27. $6x_1 + 11x_2 \geq 9$
28. $6x_1 + 11x_3 \geq 19$
29. $14x_0^2 + 11x_2^2 \geq 14$
30. $2x_0 - 2x_2 - 8x_3 \geq 0$
31. $x_1 + 12x_3 \leq 35$
32. $2x_0^2 + x_2^2 \leq 56$
33. $2x_0^2 + x_3^2 \leq 45$
34. $x_1 + 9x_2 \leq 62$
35. $x_2^2 + x_3^2 \leq 56$
36. $2x_0 + x_1 \leq 52$
37. $x_1^2 + x_2^2 + x_3^2 \leq 54$
38. $2x_0 + x_1 + x_2 \leq 30$
39. $2x_0 + x_1 + x_3 \leq 55$
40. $2x_0 + x_1 + x_2 + x_3 \leq 55$
41. $6x_1 + 5x_2 \leq 47$
42. $12x_0^2 + 6x_1^2 \leq 27$
43. $5x_2^2 + 13x_3^2 \leq 89$
44. $12x_0^2 + 6x_1^2 + 5x_2^2 \leq 52$
45. $6x_1^2 + 5x_2^2 + 13x_3^2 \leq 62$
46. $12x_0 + 6x_1 + 5x_2 + 13x_3 \leq 62$
47. $11x_2 + 11x_3 \leq 96$
48. $6x_1^2 + 11x_3^2 \leq 36$
49. $14x_0 + 11x_2 \leq 93$
50. $6x_1^2 + 11x_2^2 \leq 29$
51. $14x_0 + 6x_1 + 11x_2 + 11x_3 \leq 29$

## Step 4: Write the Gurobi code
```python
import gurobi

# Define the model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by John
x1 = m.addVar(name="x1", lb=0)  # hours worked by Dale
x2 = m.addVar(name="x2", lb=0)  # hours worked by Mary
x3 = m.addVar(name="x3", lb=0)  # hours worked by Bobby

# Define the objective function
m.setObjective(4*x0**2 + 9*x0*x1 + 8*x0*x2 + 9*x0*x3 + 2*x1**2 + 3*x1*x2 + 5*x1*x3 + 4*x2**2 + x2*x3 + 6*x3**2 + 5*x0 + 6*x1 + x2 + 9*x3, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(2*x0 <= 78)
m.addConstr(12*x0 <= 110)
m.addConstr(14*x0 <= 99)
m.addConstr(x1 <= 78)
m.addConstr(6*x1 <= 110)
m.addConstr(6*x1 <= 99)
m.addConstr(9*x2 <= 78)
m.addConstr(5*x2 <= 110)
m.addConstr(11*x2 <= 99)
m.addConstr(12*x3 <= 78)
m.addConstr(13*x3 <= 110)
m.addConstr(11*x3 <= 99)
m.addConstr(2*x0**2 + x1**2 + x2**2 >= 10)
m.addConstr(2*x0 + x1 + x3 >= 10)
m.addConstr(x1 + x2 + x3 >= 10)
m.addConstr(2*x0 + x1 + x2 >= 12)
m.addConstr(2*x0 + x1 + x3 >= 12)
m.addConstr(x1**2 + x2**2 + x3**2 >= 12)
m.addConstr(2*x0**2 + x1**2 + x2**2 >= 11)
m.addConstr(2*x0**2 + x1**2 + x3**2 >= 11)
m.addConstr(x1 + x2 + x3 >= 11)
m.addConstr(12*x0 + 6*x1 >= 14)
m.addConstr(12*x0 + 5*x2 >= 15)
m.addConstr(5*x2 + 13*x3 >= 17)
m.addConstr(12*x0 + 13*x3 >= 15)
m.addConstr(14*x0**2 + 6*x1**2 >= 11)
m.addConstr(6*x1 + 11*x2 >= 9)
m.addConstr(6*x1 + 11*x3 >= 19)
m.addConstr(14*x0**2 + 11*x2**2 >= 14)
m.addConstr(2*x0 - 2*x2 - 8*x3 >= 0)
m.addConstr(x1 + 12*x3 <= 35)
m.addConstr(2*x0**2 + x2**2 <= 56)
m.addConstr(2*x0**2 + x3**2 <= 45)
m.addConstr(x1 + 9*x2 <= 62)
m.addConstr(x2**2 + x3**2 <= 56)
m.addConstr(2*x0 + x1 <= 52)
m.addConstr(x1**2 + x2**2 + x3**2 <= 54)
m.addConstr(2*x0 + x1 + x2 <= 30)
m.addConstr(2*x0 + x1 + x3 <= 55)
m.addConstr(2*x0 + x1 + x2 + x3 <= 55)
m.addConstr(6*x1 + 5*x2 <= 47)
m.addConstr(12*x0**2 + 6*x1**2 <= 27)
m.addConstr(5*x2**2 + 13*x3**2 <= 89)
m.addConstr(12*x0**2 + 6*x1**2 + 5*x2**2 <= 52)
m.addConstr(6*x1**2 + 5*x2**2 + 13*x3**2 <= 62)
m.addConstr(12*x0 + 6*x1 + 5*x2 + 13*x3 <= 62)
m.addConstr(11*x2 + 11*x3 <= 96)
m.addConstr(6*x1**2 + 11*x3**2 <= 36)
m.addConstr(14*x0 + 11*x2 <= 93)
m.addConstr(6*x1**2 + 11*x2**2 <= 29)
m.addConstr(14*x0 + 6*x1 + 11*x2 + 11*x3 <= 29)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
else:
    print("The model is infeasible")
```

## Step 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by John'),
        ('x1', 'hours worked by Dale'),
        ('x2', 'hours worked by Mary'),
        ('x3', 'hours worked by Bobby')
    ],
    'objective_function': '4*x0^2 + 9*x0*x1 + 8*x0*x2 + 9*x0*x3 + 2*x1^2 + 3*x1*x2 + 5*x1*x3 + 4*x2^2 + x2*x3 + 6*x3^2 + 5*x0 + 6*x1 + x2 + 9*x3',
    'constraints': [
        '2*x0 <= 78',
        '12*x0 <= 110',
        '14*x0 <= 99',
        'x1 <= 78',
        '6*x1 <= 110',
        '6*x1 <= 99',
        '9*x2 <= 78',
        '5*x2 <= 110',
        '11*x2 <= 99',
        '12*x3 <= 78',
        '13*x3 <= 110',
        '11*x3 <= 99',
        '2*x0^2 + x1^2 + x2^2 >= 10',
        '2*x0 + x1 + x3 >= 10',
        'x1 + x2 + x3 >= 10',
        '2*x0 + x1 + x2 >= 12',
        '2*x0 + x1 + x3 >= 12',
        'x1^2 + x2^2 + x3^2 >= 12',
        '2*x0^2 + x1^2 + x2^2 >= 11',
        '2*x0^2 + x1^2 + x3^2 >= 11',
        'x1 + x2 + x3 >= 11',
        '12*x0 + 6*x1 >= 14',
        '12*x0 + 5*x2 >= 15',
        '5*x2 + 13*x3 >= 17',
        '12*x0 + 13*x3 >= 15',
        '14*x0^2 + 6*x1^2 >= 11',
        '6*x1 + 11*x2 >= 9',
        '6*x1 + 11*x3 >= 19',
        '14*x0^2 + 11*x2^2 >= 14',
        '2*x0 - 2*x2 - 8*x3 >= 0',
        'x1 + 12*x3 <= 35',
        '2*x0^2 + x2^2 <= 56',
        '2*x0^2 + x3^2 <= 45',
        'x1 + 9*x2 <= 62',
        'x2^2 + x3^2 <= 56',
        '2*x0 + x1 <= 52',
        'x1^2 + x2^2 + x3^2 <= 54',
        '2*x0 + x1 + x2 <= 30',
        '2*x0 + x1 + x3 <= 55',
        '2*x0 + x1 + x2 + x3 <= 55',
        '6*x1 + 5*x2 <= 47',
        '12*x0^2 + 6*x1^2 <= 27',
        '5*x2^2 + 13*x3^2 <= 89',
        '12*x0^2 + 6*x1^2 + 5*x2^2 <= 52',
        '6*x1^2 + 5*x2^2 + 13*x3^2 <= 62',
        '12*x0 + 6*x1 + 5*x2 + 13*x3 <= 62',
        '11*x2 + 11*x3 <= 96',
        '6*x1^2 + 11*x3^2 <= 36',
        '14*x0 + 11*x2 <= 93',
        '6*x1^2 + 11*x2^2 <= 29',
        '14*x0 + 6*x1 + 11*x2 + 11*x3 <= 29'
    ]
}
```