## Step 1: Define the symbolic variables and their corresponding natural language objects.
The symbolic variables and their corresponding natural language objects are:
- $x_0$ : hours worked by Laura
- $x_1$ : hours worked by John
- $x_2$ : hours worked by Paul
- $x_3$ : hours worked by Hank

## Step 2: Translate the objective function into algebraic terms using symbolic variables.
The objective function to minimize is:
$7.27x_0^2 + 2.53x_0x_2 + 9.86x_0x_3 + 7.41x_1x_3 + 6.01x_2^2 + 5.57x_2x_3 + 6.29x_3^2 + 5.81x_0 + 9.66x_1 + 9.75x_2$

## 3: List the constraints in semi-algebraic form using symbolic variables.
Constraints:
1. $11x_0 \leq 231$
2. $9x_0 \leq 264$
3. $12x_0 \leq 239$
4. $8x_1 \leq 231$
5. $19x_1 \leq 264$
6. $12x_1 \leq 239$
7. $6x_2 \leq 231$
8. $18x_2 \leq 264$
9. $6x_2 \leq 239$
10. $11x_3 \leq 231$
11. $5x_3 \leq 264$
12. $13x_3 \leq 239$
13. $11x_0^2 + 11x_3^2 \geq 54$
14. $11x_0 + 6x_2 \geq 52$
15. $8x_1^2 + 6x_2^2 \geq 53$
16. $8x_1 + 11x_3 \geq 40$
17. $11x_0 + 8x_1 + 6x_2 + 11x_3 \geq 40$
18. $9x_0 + 19x_1 \geq 47$
19. $18x_2 + 5x_3 \geq 56$
20. $9x_0 + 19x_1 + 18x_2 + 5x_3 \geq 56$
21. $12x_1 + 13x_3 \geq 55$
22. $12x_0^2 + 12x_1^2 \geq 48$
23. $12x_0 + 12x_1 + 6x_2 + 13x_3 \geq 48$
24. $6x_0^2 - 9x_1^2 \geq 0$
25. $11x_0 + 6x_2 + 11x_3 \leq 161$
26. $11x_0 + 8x_1 + 6x_2 \leq 197$
27. $19x_1 + 18x_2 \leq 250$
28. $18x_2^2 + 5x_3^2 \leq 112$
29. $19x_1 + 18x_2 + 5x_3 \leq 102$
30. $9x_0^2 + 19x_1^2 + 18x_2^2 \leq 144$
31. $12x_1 + 6x_2 \leq 152$
32. $12x_0 + 12x_1 + 6x_2 \leq 215$
33. $12x_0 + 12x_1 + 13x_3 \leq 180$
34. $12x_1^2 + 6x_2^2 + 13x_3^2 \leq 220$

## 4: Determine the variable types.
- $x_0$ : Continuous
- $x_1$ : Integer
- $x_2$ : Integer
- $x_3$ : Continuous

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

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

    # Define variables
    x0 = model.addVar(lb=0, name="hours_worked_by_Laura", vtype=gurobi.GRB.CONTINUOUS)
    x1 = model.addVar(lb=0, name="hours_worked_by_John", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(lb=0, name="hours_worked_by_Paul", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(lb=0, name="hours_worked_by_Hank", vtype=gurobi.GRB.CONTINUOUS)

    # Objective function
    model.setObjective(7.27*x0**2 + 2.53*x0*x2 + 9.86*x0*x3 + 7.41*x1*x3 + 6.01*x2**2 + 5.57*x2*x3 + 6.29*x3**2 + 5.81*x0 + 9.66*x1 + 9.75*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(11*x0 <= 231)
    model.addConstr(9*x0 <= 264)
    model.addConstr(12*x0 <= 239)
    model.addConstr(8*x1 <= 231)
    model.addConstr(19*x1 <= 264)
    model.addConstr(12*x1 <= 239)
    model.addConstr(6*x2 <= 231)
    model.addConstr(18*x2 <= 264)
    model.addConstr(6*x2 <= 239)
    model.addConstr(11*x3 <= 231)
    model.addConstr(5*x3 <= 264)
    model.addConstr(13*x3 <= 239)
    model.addConstr(11*x0**2 + 11*x3**2 >= 54)
    model.addConstr(11*x0 + 6*x2 >= 52)
    model.addConstr(8*x1**2 + 6*x2**2 >= 53)
    model.addConstr(8*x1 + 11*x3 >= 40)
    model.addConstr(11*x0 + 8*x1 + 6*x2 + 11*x3 >= 40)
    model.addConstr(9*x0 + 19*x1 >= 47)
    model.addConstr(18*x2 + 5*x3 >= 56)
    model.addConstr(9*x0 + 19*x1 + 18*x2 + 5*x3 >= 56)
    model.addConstr(12*x1 + 13*x3 >= 55)
    model.addConstr(12*x0**2 + 12*x1**2 >= 48)
    model.addConstr(12*x0 + 12*x1 + 6*x2 + 13*x3 >= 48)
    model.addConstr(6*x0**2 - 9*x1**2 >= 0)
    model.addConstr(11*x0 + 6*x2 + 11*x3 <= 161)
    model.addConstr(11*x0 + 8*x1 + 6*x2 <= 197)
    model.addConstr(19*x1 + 18*x2 <= 250)
    model.addConstr(18*x2**2 + 5*x3**2 <= 112)
    model.addConstr(19*x1 + 18*x2 + 5*x3 <= 102)
    model.addConstr(9*x0**2 + 19*x1**2 + 18*x2**2 <= 144)
    model.addConstr(12*x1 + 6*x2 <= 152)
    model.addConstr(12*x0 + 12*x1 + 6*x2 <= 215)
    model.addConstr(12*x0 + 12*x1 + 13*x3 <= 180)
    model.addConstr(12*x1**2 + 6*x2**2 + 13*x3**2 <= 220)

    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print("Hours worked by Laura: ", x0.varValue)
        print("Hours worked by John: ", x1.varValue)
        print("Hours worked by Paul: ", x2.varValue)
        print("Hours worked by Hank: ", x3.varValue)
    else:
        print("No optimal solution found.")

optimization_problem()
```

## Step 6: Provide the symbolic representation of the problem.
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Laura'), 
        ('x1', 'hours worked by John'), 
        ('x2', 'hours worked by Paul'), 
        ('x3', 'hours worked by Hank')
    ], 
    'objective_function': '7.27*x0^2 + 2.53*x0*x2 + 9.86*x0*x3 + 7.41*x1*x3 + 6.01*x2^2 + 5.57*x2*x3 + 6.29*x3^2 + 5.81*x0 + 9.66*x1 + 9.75*x2', 
    'constraints': [
        '11*x0 <= 231',
        '9*x0 <= 264',
        '12*x0 <= 239',
        '8*x1 <= 231',
        '19*x1 <= 264',
        '12*x1 <= 239',
        '6*x2 <= 231',
        '18*x2 <= 264',
        '6*x2 <= 239',
        '11*x3 <= 231',
        '5*x3 <= 264',
        '13*x3 <= 239',
        '11*x0^2 + 11*x3^2 >= 54',
        '11*x0 + 6*x2 >= 52',
        '8*x1^2 + 6*x2^2 >= 53',
        '8*x1 + 11*x3 >= 40',
        '11*x0 + 8*x1 + 6*x2 + 11*x3 >= 40',
        '9*x0 + 19*x1 >= 47',
        '18*x2 + 5*x3 >= 56',
        '9*x0 + 19*x1 + 18*x2 + 5*x3 >= 56',
        '12*x1 + 13*x3 >= 55',
        '12*x0^2 + 12*x1^2 >= 48',
        '12*x0 + 12*x1 + 6*x2 + 13*x3 >= 48',
        '6*x0^2 - 9*x1^2 >= 0',
        '11*x0 + 6*x2 + 11*x3 <= 161',
        '11*x0 + 8*x1 + 6*x2 <= 197',
        '19*x1 + 18*x2 <= 250',
        '18*x2^2 + 5*x3^2 <= 112',
        '19*x1 + 18*x2 + 5*x3 <= 102',
        '9*x0^2 + 19*x1^2 + 18*x2^2 <= 144',
        '12*x1 + 6*x2 <= 152',
        '12*x0 + 12*x1 + 6*x2 <= 215',
        '12*x0 + 12*x1 + 13*x3 <= 180',
        '12*x1^2 + 6*x2^2 + 13*x3^2 <= 220'
    ]
}
```