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

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $5.02x_1^2 + 4.37x_1x_2 + 3.47x_2^2 + 3.78x_2$.

## Step 3: List the constraints in symbolic notation
The constraints are:
1. $x_1 \geq 0$ and $x_2 \geq 0$ (non-negativity, but hours worked by Bill cannot be a fraction, so $x_1$ must be an integer)
2. $8x_1^2 + 13x_2^2 \geq 35$
3. $2x_1 + 9x_2 \geq 41$
4. $6x_1 + 12x_2 \geq 38$
5. $3x_1 - 6x_2 \geq 0$
6. $8x_1^2 + 13x_2^2 \leq 123$
7. $8x_1 + 13x_2 \leq 123$
8. $2x_1^2 + 9x_2^2 \leq 80$
9. $2x_1 + 9x_2 \leq 80$
10. $6x_1 + 12x_2 \leq 71$
11. $x_1$ is an integer.

## 4: Convert the problem into a Gurobi-compatible format
We will use Gurobi's Python API to model and solve this problem.

## 5: Write the Gurobi code
```python
import gurobi as gp
from gurobi import GRB

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_Bill", vtype=GRB.INTEGER)  # Bill's hours
x2 = m.addVar(name="hours_worked_by_John")  # John's hours

# Define the objective function
m.setObjective(5.02*x1**2 + 4.37*x1*x2 + 3.47*x2**2 + 3.78*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(8*x1**2 + 13*x2**2 >= 35, name="dollar_cost_constraint")
m.addConstr(2*x1 + 9*x2 >= 41, name="work_quality_constraint")
m.addConstr(6*x1 + 12*x2 >= 38, name="likelihood_to_quit_constraint")
m.addConstr(3*x1 - 6*x2 >= 0, name="hours_worked_constraint")
m.addConstr(8*x1**2 + 13*x2**2 <= 123, name="dollar_cost_upper_bound")
m.addConstr(8*x1 + 13*x2 <= 123, name="total_dollar_cost")
m.addConstr(2*x1**2 + 9*x2**2 <= 80, name="work_quality_upper_bound")
m.addConstr(2*x1 + 9*x2 <= 80, name="total_work_quality")
m.addConstr(6*x1 + 12*x2 <= 71, name="total_likelihood_to_quit")

# Optimize the model
m.optimize()

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

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Bill'), ('x2', 'hours worked by John')],
    'objective_function': '5.02*x1^2 + 4.37*x1*x2 + 3.47*x2^2 + 3.78*x2',
    'constraints': [
        '8*x1^2 + 13*x2^2 >= 35',
        '2*x1 + 9*x2 >= 41',
        '6*x1 + 12*x2 >= 38',
        '3*x1 - 6*x2 >= 0',
        '8*x1^2 + 13*x2^2 <= 123',
        '8*x1 + 13*x2 <= 123',
        '2*x1^2 + 9*x2^2 <= 80',
        '2*x1 + 9*x2 <= 80',
        '6*x1 + 12*x2 <= 71',
        'x1 is an integer'
    ]
}
```