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

## Step 2: Convert the natural language objects into symbolic notation
The symbolic representation of the variables and their corresponding natural language objects are:
- $x_0$ : 'hours worked by Dale'
- $x_1$ : 'hours worked by Bobby'
- $x_2$ : 'hours worked by Jean'

## 3: Define the objective function in symbolic notation
The objective function to maximize is $7.88x_0^2 + 3.37x_0x_2 + 5.46x_1x_2 + 4.27x_2^2 + 7.6x_0 + 3.89x_1 + 7.16x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $18x_0 \leq 261$
- $12x_0 \leq 159$
- $22x_1 \leq 261$
- $27x_1 \leq 159$
- $9x_2 \leq 261$
- $13x_2 \leq 159$
- $18x_0 + 22x_1 \geq 51$
- $22^2x_1^2 + 9^2x_2^2 \geq 38$
- $12x_0 + 27x_1 \geq 20$
- $18x_0 + 22x_1 \leq 236$
- $18x_0 + 22x_1 + 9x_2 \leq 236$
- $12x_0 + 27x_1 \leq 135$
- $27x_1 + 13x_2 \leq 144$
- $12x_0 + 27x_1 + 13x_2 \leq 144$

## 5: Create the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Dale'), 
        ('x1', 'hours worked by Bobby'), 
        ('x2', 'hours worked by Jean')
    ], 
    'objective_function': '7.88*x0^2 + 3.37*x0*x2 + 5.46*x1*x2 + 4.27*x2^2 + 7.6*x0 + 3.89*x1 + 7.16*x2', 
    'constraints': [
        '18*x0 <= 261', 
        '12*x0 <= 159', 
        '22*x1 <= 261', 
        '27*x1 <= 159', 
        '9*x2 <= 261', 
        '13*x2 <= 159', 
        '18*x0 + 22*x1 >= 51', 
        '22^2*x1^2 + 9^2*x2^2 >= 38', 
        '12*x0 + 27*x1 >= 20', 
        '18*x0 + 22*x1 <= 236', 
        '18*x0 + 22*x1 + 9*x2 <= 236', 
        '12*x0 + 27*x1 <= 135', 
        '27*x1 + 13*x2 <= 144', 
        '12*x0 + 27*x1 + 13*x2 <= 144'
    ]
}
```

## Step 6: Write the Gurobi code to solve the optimization problem
```python
import gurobi

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define the variables
    x0 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Dale')
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Bobby')
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Jean')

    # Define the objective function
    model.setObjective(7.88*x0**2 + 3.37*x0*x2 + 5.46*x1*x2 + 4.27*x2**2 + 7.6*x0 + 3.89*x1 + 7.16*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(18*x0 <= 261)
    model.addConstr(12*x0 <= 159)
    model.addConstr(22*x1 <= 261)
    model.addConstr(27*x1 <= 159)
    model.addConstr(9*x2 <= 261)
    model.addConstr(13*x2 <= 159)
    model.addConstr(18*x0 + 22*x1 >= 51)
    model.addConstr(22**2*x1**2 + 9**2*x2**2 >= 38)
    model.addConstr(12*x0 + 27*x1 >= 20)
    model.addConstr(18*x0 + 22*x1 <= 236)
    model.addConstr(18*x0 + 22*x1 + 9*x2 <= 236)
    model.addConstr(12*x0 + 27*x1 <= 135)
    model.addConstr(27*x1 + 13*x2 <= 144)
    model.addConstr(12*x0 + 27*x1 + 13*x2 <= 144)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print('Objective: ', model.objVal)
        print('Hours worked by Dale: ', x0.x)
        print('Hours worked by Bobby: ', x1.x)
        print('Hours worked by Jean: ', x2.x)
    else:
        print('The model is infeasible')

solve_optimization_problem()
```