## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill' and 'hours worked by Mary', which we can denote as $x_0$ and $x_1$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $8.3x_0^2 + 5.41x_0x_1 + 3.95x_1^2 + 8.68x_0 + 2.57x_1$.

## Step 3: List the constraints
The constraints are:
- $x_0 \geq 0$ and $x_1 \geq 0$ (non-negativity constraints, but $x_1$ must be an integer)
- $8x_0 \leq 46$ and $6x_1 \leq 46$ (likelihood to quit index constraints for Bill and Mary)
- $17x_0 \leq 144$ and $7x_1 \leq 144$ (productivity rating constraints for Bill and Mary)
- $3x_0 \leq 71$ and $11x_1 \leq 71$ (computer competence rating constraints for Bill and Mary)
- $5x_0 \leq 91$ and $11x_1 \leq 91$ (organization score constraints for Bill and Mary)
- $5x_0 \leq 92$ and $5x_1 \leq 92$ (paperwork competence rating constraints for Bill and Mary)
- $8x_0 + 6x_1 \geq 15$ (total combined likelihood to quit index constraint)
- $17x_0 + 7x_1 \geq 45$ (total combined productivity rating constraint)
- $3x_0 + 11x_1 \geq 20$ (total combined computer competence rating constraint)
- $5x_0 + 11x_1 \geq 27$ (total combined organization score constraint)
- $5x_0^2 + 5x_1^2 \geq 46$ (total combined paperwork competence rating constraint)
- $8x_0^2 - 4x_1^2 \geq 0$ (additional constraint)
- $8x_0 + 6x_1 \leq 28$ (total combined likelihood to quit index upper bound constraint)
- $17x_0 + 7x_1 \leq 119$ (total combined productivity rating upper bound constraint)
- $3x_0 + 11x_1 \leq 42$ (total combined computer competence rating upper bound constraint)
- $5x_0 + 11x_1 \leq 69$ (total combined organization score upper bound constraint)
- $5x_0 + 5x_1 \leq 59$ (total combined paperwork competence rating upper bound constraint)

## Step 4: Convert the problem into Gurobi code
We will use Gurobi's Python API to model and solve this problem.

```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="hours_worked_by_Bill", lb=0)  # Bill's hours
x1 = m.addVar(name="hours_worked_by_Mary", lb=0, vtype=gp.GRB.INTEGER)  # Mary's hours

# Define the objective function
m.setObjective(8.3*x0**2 + 5.41*x0*x1 + 3.95*x1**2 + 8.68*x0 + 2.57*x1, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(8*x0 <= 46)
m.addConstr(6*x1 <= 46)
m.addConstr(17*x0 <= 144)
m.addConstr(7*x1 <= 144)
m.addConstr(3*x0 <= 71)
m.addConstr(11*x1 <= 71)
m.addConstr(5*x0 <= 91)
m.addConstr(11*x1 <= 91)
m.addConstr(5*x0 <= 92)
m.addConstr(5*x1 <= 92)
m.addConstr(8*x0 + 6*x1 >= 15)
m.addConstr(17*x0 + 7*x1 >= 45)
m.addConstr(3*x0 + 11*x1 >= 20)
m.addConstr(5*x0 + 11*x1 >= 27)
m.addConstr(5*x0**2 + 5*x1**2 >= 46)
m.addConstr(8*x0**2 - 4*x1**2 >= 0)
m.addConstr(8*x0 + 6*x1 <= 28)
m.addConstr(17*x0 + 7*x1 <= 119)
m.addConstr(3*x0 + 11*x1 <= 42)
m.addConstr(5*x0 + 11*x1 <= 69)
m.addConstr(5*x0 + 5*x1 <= 59)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Bill: {x0.varValue}")
    print(f"Hours worked by Mary: {x1.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## Step 5: Symbolic representation of the problem
The symbolic representation is as follows:

```json
{
    'sym_variables': [('x0', 'hours worked by Bill'), ('x1', 'hours worked by Mary')],
    'objective_function': '8.3*x0^2 + 5.41*x0*x1 + 3.95*x1^2 + 8.68*x0 + 2.57*x1',
    'constraints': [
        '8*x0 <= 46',
        '6*x1 <= 46',
        '17*x0 <= 144',
        '7*x1 <= 144',
        '3*x0 <= 71',
        '11*x1 <= 71',
        '5*x0 <= 91',
        '11*x1 <= 91',
        '5*x0 <= 92',
        '5*x1 <= 92',
        '8*x0 + 6*x1 >= 15',
        '17*x0 + 7*x1 >= 45',
        '3*x0 + 11*x1 >= 20',
        '5*x0 + 11*x1 >= 27',
        '5*x0^2 + 5*x1^2 >= 46',
        '8*x0^2 - 4*x1^2 >= 0',
        '8*x0 + 6*x1 <= 28',
        '17*x0 + 7*x1 <= 119',
        '3*x0 + 11*x1 <= 42',
        '5*x0 + 11*x1 <= 69',
        '5*x0 + 5*x1 <= 59'
    ]
}
```