## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by John', 'hours worked by Jean', 'hours worked by Peggy', 'hours worked by George'] which can be represented symbolically as 'x1', 'x2', 'x3', 'x4' respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is: $1.99x_1^2 + 8.52x_1x_2 + 3.5x_1x_3 + 8.82x_1x_4 + 4.38x_2^2 + 5.59x_2x_4 + 4.65x_3^2 + 3.94x_3x_4 + 8.1x_4^2 + 3.24x_1 + 3.63x_2 + 8.56x_3$.

## 3: List the constraints in symbolic notation
The constraints are:
- $3x_1 \leq 77$
- $11x_2 \leq 77$
- $13x_3 \leq 77$
- $1x_4 \leq 77$
- $13x_3 + 1x_4 \geq 6$
- $3x_1 + 13x_3 \geq 6$
- $3x_1 + 11x_2 + 13x_3 \geq 13$
- $3^2x_1^2 + 13^2x_3^2 + 1^2x_4^2 \geq 13$
- $11x_2 + 13x_3 + 1x_4 \geq 13$
- $3x_1 + 13x_3 + 1x_4 \geq 13$
- $11x_2 + 13x_3 + 1x_4 \geq 13$
- $3x_1 + 11x_2 + 13x_3 \geq 10$
- $3x_1 + 13x_3 + 1x_4 \geq 10$
- $11x_2 + 13x_3 + 1x_4 \geq 10$
- $13x_3 + 1x_4 \leq 42$
- $3x_1 + 13x_3 + 1x_4 \leq 58$
- $3^2x_1^2 + 11^2x_2^2 + 1^2x_4^2 \leq 66$
- $3x_1 + 11x_2 + 13x_3 + 1x_4 \leq 66$
- $x_1 \in \mathbb{Z}$ (integer constraint for $x_1$)

## 4: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [
        ('x1', 'hours worked by John'), 
        ('x2', 'hours worked by Jean'), 
        ('x3', 'hours worked by Peggy'), 
        ('x4', 'hours worked by George')
    ], 
    'objective_function': '1.99*x1^2 + 8.52*x1*x2 + 3.5*x1*x3 + 8.82*x1*x4 + 4.38*x2^2 + 5.59*x2*x4 + 4.65*x3^2 + 3.94*x3*x4 + 8.1*x4^2 + 3.24*x1 + 3.63*x2 + 8.56*x3',
    'constraints': [
        '3*x1 <= 77', 
        '11*x2 <= 77', 
        '13*x3 <= 77', 
        '1*x4 <= 77', 
        '13*x3 + 1*x4 >= 6', 
        '3*x1 + 13*x3 >= 6', 
        '3*x1 + 11*x2 + 13*x3 >= 13', 
        '3^2*x1^2 + 13^2*x3^2 + 1^2*x4^2 >= 13', 
        '11*x2 + 13*x3 + 1*x4 >= 13', 
        '3*x1 + 13*x3 + 1*x4 >= 13', 
        '11*x2 + 13*x3 + 1*x4 >= 13', 
        '3*x1 + 11*x2 + 13*x3 >= 10', 
        '3*x1 + 13*x3 + 1*x4 >= 10', 
        '11*x2 + 13*x3 + 1*x4 >= 10', 
        '13*x3 + 1*x4 <= 42', 
        '3*x1 + 13*x3 + 1*x4 <= 58', 
        '3^2*x1^2 + 11^2*x2^2 + 1^2*x4^2 <= 66', 
        '3*x1 + 11*x2 + 13*x3 + 1*x4 <= 66'
    ]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi as gp

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_John", vtype=gp.GRB.INTEGER)  # integer constraint for x1
x2 = m.addVar(name="hours_worked_by_Jean")
x3 = m.addVar(name="hours_worked_by_Peggy")
x4 = m.addVar(name="hours_worked_by_George")

# Objective function
m.setObjective(1.99*x1**2 + 8.52*x1*x2 + 3.5*x1*x3 + 8.82*x1*x4 + 4.38*x2**2 + 5.59*x2*x4 + 4.65*x3**2 + 3.94*x3*x4 + 8.1*x4**2 + 3.24*x1 + 3.63*x2 + 8.56*x3, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(3*x1 <= 77)
m.addConstr(11*x2 <= 77)
m.addConstr(13*x3 <= 77)
m.addConstr(1*x4 <= 77)
m.addConstr(13*x3 + 1*x4 >= 6)
m.addConstr(3*x1 + 13*x3 >= 6)
m.addConstr(3*x1 + 11*x2 + 13*x3 >= 13)
m.addConstr(3**2*x1**2 + 13**2*x3**2 + 1**2*x4**2 >= 13)
m.addConstr(11*x2 + 13*x3 + 1*x4 >= 13)
m.addConstr(3*x1 + 13*x3 + 1*x4 >= 13)
m.addConstr(11*x2 + 13*x3 + 1*x4 >= 13)
m.addConstr(3*x1 + 11*x2 + 13*x3 >= 10)
m.addConstr(3*x1 + 13*x3 + 1*x4 >= 10)
m.addConstr(11*x2 + 13*x3 + 1*x4 >= 10)
m.addConstr(13*x3 + 1*x4 <= 42)
m.addConstr(3*x1 + 13*x3 + 1*x4 <= 58)
m.addConstr(3**2*x1**2 + 11**2*x2**2 + 1**2*x4**2 <= 66)
m.addConstr(3*x1 + 11*x2 + 13*x3 + 1*x4 <= 66)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by John: {x1.varValue}")
    print(f"Hours worked by Jean: {x2.varValue}")
    print(f"Hours worked by Peggy: {x3.varValue}")
    print(f"Hours worked by George: {x4.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```