## Step 1: Define the symbolic representation of the problem
The problem involves maximizing an objective function subject to several constraints. The variables are 'hours worked by Ringo', 'hours worked by Bobby', 'hours worked by Dale', and 'hours worked by Laura', which we will denote as $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to maximize is $9x_1^2 + 6x_1x_2 + 5x_1x_4 + 4x_2x_4 + 9x_3x_4 + x_3 + 8x_4$.

## 3: List the constraints in symbolic notation
The constraints are:
- $1.38x_1 \leq 322$
- $18.15x_1 \leq 440$
- $6.59x_2 \leq 322$
- $8.98x_2 \leq 440$
- $18.38x_3 \leq 322$
- $13.32x_3 \leq 440$
- $1.66x_4 \leq 322$
- $21.23x_4 \leq 440$
- $1.38x_1 + 1.66x_4 \geq 28$
- $6.59x_2 + 1.66x_4 \geq 71$
- $18.38x_3 + 1.66x_4 \geq 31$
- $1.38^2x_1^2 + 18.38^2x_3^2 \geq 42$
- $1.38^2x_1^2 + 18.38^2x_3^2 + 1.66^2x_4^2 \geq 69$
- $8.98x_2 + 13.32x_3 \geq 79$
- $13.32x_3 + 21.23x_4 \geq 45$
- $18.15x_1 + 8.98x_2 + 21.23x_4 \geq 102$
- $8.98^2x_2^2 + 13.32^2x_3^2 + 21.23^2x_4^2 \geq 102$
- $18.15x_1 + 8.98x_2 + 21.23x_4 \geq 81$
- $8.98^2x_2^2 + 13.32^2x_3^2 + 21.23^2x_4^2 \geq 81$
- $1.38x_1 + 18.38x_3 \leq 298$
- $6.59^2x_2^2 + 18.38^2x_3^2 \leq 220$
- $1.38x_1 + 1.66x_4 \leq 169$
- $1.38^2x_1^2 + 6.59^2x_2^2 \leq 144$
- $6.59x_2 + 1.66x_4 \leq 93$
- $18.38x_3 + 1.66x_4 \leq 149$
- $1.38x_1 + 6.59x_2 + 18.38x_3 + 1.66x_4 \leq 149$
- $18.15^2x_1^2 + 21.23^2x_4^2 \leq 361$
- $13.32x_3 + 21.23x_4 \leq 316$
- $18.15^2x_1^2 + 8.98^2x_2^2 \leq 126$
- $18.15x_1 + 13.32x_3 + 21.23x_4 \leq 435$
- $18.15^2x_1^2 + 8.98^2x_2^2 + 21.23^2x_4^2 \leq 297$
- $18.15x_1 + 8.98x_2 + 13.32x_3 + 21.23x_4 \leq 297$

## 4: Define the symbolic variables and constraints for the JSON output
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Ringo'),
        ('x2', 'hours worked by Bobby'),
        ('x3', 'hours worked by Dale'),
        ('x4', 'hours worked by Laura')
    ],
    'objective_function': '9*x1^2 + 6*x1*x2 + 5*x1*x4 + 4*x2*x4 + 9*x3*x4 + x3 + 8*x4',
    'constraints': [
        '1.38*x1 <= 322',
        '18.15*x1 <= 440',
        '6.59*x2 <= 322',
        '8.98*x2 <= 440',
        '18.38*x3 <= 322',
        '13.32*x3 <= 440',
        '1.66*x4 <= 322',
        '21.23*x4 <= 440',
        '1.38*x1 + 1.66*x4 >= 28',
        '6.59*x2 + 1.66*x4 >= 71',
        '18.38*x3 + 1.66*x4 >= 31',
        '1.38^2*x1^2 + 18.38^2*x3^2 >= 42',
        '1.38^2*x1^2 + 18.38^2*x3^2 + 1.66^2*x4^2 >= 69',
        '8.98*x2 + 13.32*x3 >= 79',
        '13.32*x3 + 21.23*x4 >= 45',
        '18.15*x1 + 8.98*x2 + 21.23*x4 >= 102',
        '8.98^2*x2^2 + 13.32^2*x3^2 + 21.23^2*x4^2 >= 102',
        '18.15*x1 + 8.98*x2 + 21.23*x4 >= 81',
        '8.98^2*x2^2 + 13.32^2*x3^2 + 21.23^2*x4^2 >= 81',
        '1.38*x1 + 18.38*x3 <= 298',
        '6.59^2*x2^2 + 18.38^2*x3^2 <= 220',
        '1.38*x1 + 1.66*x4 <= 169',
        '1.38^2*x1^2 + 6.59^2*x2^2 <= 144',
        '6.59*x2 + 1.66*x4 <= 93',
        '18.38*x3 + 1.66*x4 <= 149',
        '1.38*x1 + 6.59*x2 + 18.38*x3 + 1.66*x4 <= 149',
        '18.15^2*x1^2 + 21.23^2*x4^2 <= 361',
        '13.32*x3 + 21.23*x4 <= 316',
        '18.15^2*x1^2 + 8.98^2*x2^2 <= 126',
        '18.15*x1 + 13.32*x3 + 21.23*x4 <= 435',
        '18.15^2*x1^2 + 8.98^2*x2^2 + 21.23^2*x4^2 <= 297',
        '18.15*x1 + 8.98*x2 + 13.32*x3 + 21.23*x4 <= 297'
    ]
}
```

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

# Define the model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(name="x1", lb=-float('inf'), ub=float('inf'))  # hours worked by Ringo
x2 = m.addVar(name="x2", lb=-float('inf'), ub=float('inf'))  # hours worked by Bobby
x3 = m.addVar(name="x3", lb=0, ub=float('inf'), integrality=1)  # hours worked by Dale
x4 = m.addVar(name="x4", lb=0, ub=float('inf'), integrality=1)  # hours worked by Laura

# Define the objective function
m.setObjective(9 * x1**2 + 6 * x1 * x2 + 5 * x1 * x4 + 4 * x2 * x4 + 9 * x3 * x4 + x3 + 8 * x4, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(1.38 * x1 <= 322)
m.addConstr(18.15 * x1 <= 440)
m.addConstr(6.59 * x2 <= 322)
m.addConstr(8.98 * x2 <= 440)
m.addConstr(18.38 * x3 <= 322)
m.addConstr(13.32 * x3 <= 440)
m.addConstr(1.66 * x4 <= 322)
m.addConstr(21.23 * x4 <= 440)
m.addConstr(1.38 * x1 + 1.66 * x4 >= 28)
m.addConstr(6.59 * x2 + 1.66 * x4 >= 71)
m.addConstr(18.38 * x3 + 1.66 * x4 >= 31)
m.addConstr(1.38**2 * x1**2 + 18.38**2 * x3**2 >= 42)
m.addConstr(1.38**2 * x1**2 + 18.38**2 * x3**2 + 1.66**2 * x4**2 >= 69)
m.addConstr(8.98 * x2 + 13.32 * x3 >= 79)
m.addConstr(13.32 * x3 + 21.23 * x4 >= 45)
m.addConstr(18.15 * x1 + 8.98 * x2 + 21.23 * x4 >= 102)
m.addConstr(8.98**2 * x2**2 + 13.32**2 * x3**2 + 21.23**2 * x4**2 >= 102)
m.addConstr(18.15 * x1 + 8.98 * x2 + 21.23 * x4 >= 81)
m.addConstr(8.98**2 * x2**2 + 13.32**2 * x3**2 + 21.23**2 * x4**2 >= 81)
m.addConstr(1.38 * x1 + 18.38 * x3 <= 298)
m.addConstr(6.59**2 * x2**2 + 18.38**2 * x3**2 <= 220)
m.addConstr(1.38 * x1 + 1.66 * x4 <= 169)
m.addConstr(1.38**2 * x1**2 + 6.59**2 * x2**2 <= 144)
m.addConstr(6.59 * x2 + 1.66 * x4 <= 93)
m.addConstr(18.38 * x3 + 1.66 * x4 <= 149)
m.addConstr(1.38 * x1 + 6.59 * x2 + 18.38 * x3 + 1.66 * x4 <= 149)
m.addConstr(18.15**2 * x1**2 + 21.23**2 * x4**2 <= 361)
m.addConstr(13.32 * x3 + 21.23 * x4 <= 316)
m.addConstr(18.15**2 * x1**2 + 8.98**2 * x2**2 <= 126)
m.addConstr(18.15 * x1 + 13.32 * x3 + 21.23 * x4 <= 435)
m.addConstr(18.15**2 * x1**2 + 8.98**2 * x2**2 + 21.23**2 * x4**2 <= 297)
m.addConstr(18.15 * x1 + 8.98 * x2 + 13.32 * x3 + 21.23 * x4 <= 297)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
    print("x4: ", x4.varValue)
else:
    print("The model is infeasible")
```