## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill', 'hours worked by Ringo', 'hours worked by Bobby', and 'hours worked by Jean', which can be represented symbolically as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $8.78x_0^2 + 3.25x_0x_3 + 2.78x_2x_3 + 6.85x_1 + 5.71x_2$.

## Step 3: List the constraints in symbolic notation
Constraints:
1. $10x_0 \leq 362$
2. $15x_0 \leq 185$
3. $16x_0 \leq 374$
4. $19x_1 \leq 362$
5. $17x_1 \leq 185$
6. $21x_1 \leq 374$
7. $14x_2 \leq 362$
8. $15x_2 \leq 185$
9. $16x_2 \leq 374$
10. $10x_3 \leq 362$
11. $20x_3 \leq 185$
12. $3x_3 \leq 374$
13. $14x_2 + 10x_3 \geq 32$
14. $10x_0 + 14x_2 \geq 72$
15. $19x_1 + 14x_2 \geq 68$
16. $19x_1 + 14x_2 + 10x_3 \geq 88$
17. $10x_0 + 14x_2 + 10x_3 \geq 88$
18. $10x_0 + 19x_1 + 14x_2 \geq 88$
19. $19x_1 + 14x_2 + 10x_3 \geq 57$
20. $10x_0 + 14x_2 + 10x_3 \geq 57$
21. $10x_0 + 19x_1 + 14x_2 \geq 57$
22. $19x_1^2 + 14x_2^2 + 10x_3^2 \geq 83$
23. $10x_0^2 + 14x_2^2 + 10x_3^2 \geq 83$
24. $10x_0 + 19x_1 + 14x_2 \geq 83$
25. $10x_0 + 10x_3 \leq 283$
26. $19x_1 + 10x_3 \leq 331$
27. $19x_1 + 14x_2 \leq 105$
28. $10x_0 + 14x_2 \leq 161$
29. $10x_0^2 + 14x_2^2 + 10x_3^2 \leq 210$
30. $10x_0 + 19x_1 + 10x_3 \leq 216$
31. $19x_1^2 + 14x_2^2 + 10x_3^2 \leq 253$
32. $10x_0 + 19x_1 + 14x_2 + 10x_3 \leq 253$
33. $15x_0^2 + 17x_1^2 \leq 62$
34. $17x_1^2 + 15x_2^2 \leq 159$
35. $15x_2 + 20x_3 \leq 60$
36. $15x_0 + 15x_2 \leq 153$
37. $15x_0^2 + 17x_1^2 + 15x_2^2 \leq 146$
38. $15x_0 + 17x_1 + 15x_2 + 20x_3 \leq 146$
39. $21x_1 + 3x_3 \leq 333$
40. $21x_1^2 + 16x_2^2 \leq 149$
41. $16x_0 + 16x_2 \leq 240$
42. $16x_0 + 21x_1 + 16x_2 + 3x_3 \leq 240$

## Step 4: Create the Gurobi model and variables
We will now create a Gurobi model and define the variables.

## 5: Implement the objective function and constraints in Gurobi

```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="hours_worked_by_Bill", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x1 = m.addVar(name="hours_worked_by_Ringo", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x2 = m.addVar(name="hours_worked_by_Bobby", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x3 = m.addVar(name="hours_worked_by_Jean", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)

# Objective function
m.setObjective(8.78 * x0**2 + 3.25 * x0 * x3 + 2.78 * x2 * x3 + 6.85 * x1 + 5.71 * x2, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(10 * x0 <= 362)
m.addConstr(15 * x0 <= 185)
m.addConstr(16 * x0 <= 374)
m.addConstr(19 * x1 <= 362)
m.addConstr(17 * x1 <= 185)
m.addConstr(21 * x1 <= 374)
m.addConstr(14 * x2 <= 362)
m.addConstr(15 * x2 <= 185)
m.addConstr(16 * x2 <= 374)
m.addConstr(10 * x3 <= 362)
m.addConstr(20 * x3 <= 185)
m.addConstr(3 * x3 <= 374)
m.addConstr(14 * x2 + 10 * x3 >= 32)
m.addConstr(10 * x0 + 14 * x2 >= 72)
m.addConstr(19 * x1 + 14 * x2 >= 68)
m.addConstr(19 * x1 + 14 * x2 + 10 * x3 >= 88)
m.addConstr(10 * x0 + 14 * x2 + 10 * x3 >= 88)
m.addConstr(10 * x0 + 19 * x1 + 14 * x2 >= 88)
m.addConstr(19 * x1 + 14 * x2 + 10 * x3 >= 57)
m.addConstr(10 * x0 + 14 * x2 + 10 * x3 >= 57)
m.addConstr(10 * x0 + 19 * x1 + 14 * x2 >= 57)
m.addConstr(19 * x1**2 + 14 * x2**2 + 10 * x3**2 >= 83)
m.addConstr(10 * x0**2 + 14 * x2**2 + 10 * x3**2 >= 83)
m.addConstr(10 * x0 + 19 * x1 + 14 * x2 >= 83)
m.addConstr(10 * x0 + 10 * x3 <= 283)
m.addConstr(19 * x1 + 10 * x3 <= 331)
m.addConstr(19 * x1 + 14 * x2 <= 105)
m.addConstr(10 * x0 + 14 * x2 <= 161)
m.addConstr(10 * x0**2 + 14 * x2**2 + 10 * x3**2 <= 210)
m.addConstr(10 * x0 + 19 * x1 + 10 * x3 <= 216)
m.addConstr(19 * x1**2 + 14 * x2**2 + 10 * x3**2 <= 253)
m.addConstr(10 * x0 + 19 * x1 + 14 * x2 + 10 * x3 <= 253)
m.addConstr(15 * x0**2 + 17 * x1**2 <= 62)
m.addConstr(17 * x1**2 + 15 * x2**2 <= 159)
m.addConstr(15 * x2 + 20 * x3 <= 60)
m.addConstr(15 * x0 + 15 * x2 <= 153)
m.addConstr(15 * x0**2 + 17 * x1**2 + 15 * x2**2 <= 146)
m.addConstr(15 * x0 + 17 * x1 + 15 * x2 + 20 * x3 <= 146)
m.addConstr(21 * x1 + 3 * x3 <= 333)
m.addConstr(21 * x1**2 + 16 * x2**2 <= 149)
m.addConstr(16 * x0 + 16 * x2 <= 240)
m.addConstr(16 * x0 + 21 * x1 + 16 * x2 + 3 * x3 <= 240)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Bill: ", x0.varValue)
    print("Hours worked by Ringo: ", x1.varValue)
    print("Hours worked by Bobby: ", x2.varValue)
    print("Hours worked by Jean: ", x3.varValue)
else:
    print("The model is infeasible")
```

```json
{
    'sym_variables': [
        ['x0', 'hours worked by Bill'],
        ['x1', 'hours worked by Ringo'],
        ['x2', 'hours worked by Bobby'],
        ['x3', 'hours worked by Jean']
    ],
    'objective_function': '8.78*x0^2 + 3.25*x0*x3 + 2.78*x2*x3 + 6.85*x1 + 5.71*x2',
    'constraints': [
        '10*x0 <= 362',
        '15*x0 <= 185',
        '16*x0 <= 374',
        '19*x1 <= 362',
        '17*x1 <= 185',
        '21*x1 <= 374',
        '14*x2 <= 362',
        '15*x2 <= 185',
        '16*x2 <= 374',
        '10*x3 <= 362',
        '20*x3 <= 185',
        '3*x3 <= 374',
        '14*x2 + 10*x3 >= 32',
        '10*x0 + 14*x2 >= 72',
        '19*x1 + 14*x2 >= 68',
        '19*x1 + 14*x2 + 10*x3 >= 88',
        '10*x0 + 14*x2 + 10*x3 >= 88',
        '10*x0 + 19*x1 + 14*x2 >= 88',
        '19*x1 + 14*x2 + 10*x3 >= 57',
        '10*x0 + 14*x2 + 10*x3 >= 57',
        '10*x0 + 19*x1 + 14*x2 >= 57',
        '19*x1^2 + 14*x2^2 + 10*x3^2 >= 83',
        '10*x0^2 + 14*x2^2 + 10*x3^2 >= 83',
        '10*x0 + 19*x1 + 14*x2 >= 83',
        '10*x0 + 10*x3 <= 283',
        '19*x1 + 10*x3 <= 331',
        '19*x1 + 14*x2 <= 105',
        '10*x0 + 14*x2 <= 161',
        '10*x0^2 + 14*x2^2 + 10*x3^2 <= 210',
        '10*x0 + 19*x1 + 10*x3 <= 216',
        '19*x1^2 + 14*x2^2 + 10*x3^2 <= 253',
        '10*x0 + 19*x1 + 14*x2 + 10*x3 <= 253',
        '15*x0^2 + 17*x1^2 <= 62',
        '17*x1^2 + 15*x2^2 <= 159',
        '15*x2 + 20*x3 <= 60',
        '15*x0 + 15*x2 <= 153',
        '15*x0^2 + 17*x1^2 + 15*x2^2 <= 146',
        '15*x0 + 17*x1 + 15*x2 + 20*x3 <= 146',
        '21*x1 + 3*x3 <= 333',
        '21*x1^2 + 16*x2^2 <= 149',
        '16*x0 + 16*x2 <= 240',
        '16*x0 + 21*x1 + 16*x2 + 3*x3 <= 240'
    ]
}
```