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

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $5.86x_0^2 + 3.14x_0x_1 + 9.5x_1x_2 + 9.69x_2^2 + 6.42x_0 + 9.76x_1$.

## 3: Define the constraints in symbolic notation
The constraints are:
- $x_0 \geq 0$ (non-negativity, implicit)
- $x_1 \geq 0$ (non-negativity, implicit)
- $x_2 \geq 0$ (non-negativity, implicit)
- $x_0$ can be fractional
- $x_1$ must be integer
- $x_2$ must be integer
- $5x_0 + 1x_2 \leq 97$ (dollar cost constraint for Bill and Mary)
- $4x_1 + 1x_2 \leq 46$ (dollar cost constraint for Ringo and Mary)
- $5x_0 + 4x_1 \leq 44$ (dollar cost constraint for Bill and Ringo)
- $5x_0 + 4x_1 + 1x_2 \leq 44$ (dollar cost constraint for all)
- $9x_0^2 + 9x_2^2 \geq 29$ (work quality rating constraint for Bill and Mary)
- $9x_0^2 + 16x_1^2 \geq 21$ (work quality rating constraint for Bill and Ringo)
- $-8x_1^2 + 4x_2^2 \geq 0$ (work quality rating constraint)
- $9x_0 + 9x_2 \leq 61$ (work quality rating constraint for Bill and Mary)
- $16x_1 + 9x_2 \leq 37$ (work quality rating constraint for Ringo and Mary)
- $9x_0^2 + 16x_1^2 \leq 85$ (work quality rating constraint for Bill and Ringo)
- $9x_0 + 16x_1 + 9x_2 \leq 85$ (work quality rating constraint for all)

## 4: Create a symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Bill'), 
        ('x1', 'hours worked by Ringo'), 
        ('x2', 'hours worked by Mary')
    ], 
    'objective_function': '5.86*x0^2 + 3.14*x0*x1 + 9.5*x1*x2 + 9.69*x2^2 + 6.42*x0 + 9.76*x1', 
    'constraints': [
        '9*x0^2 + 9*x2^2 >= 29',
        '9*x0^2 + 16*x1^2 >= 21',
        '-8*x1^2 + 4*x2^2 >= 0',
        '5*x0 + 1*x2 <= 97',
        '4*x1 + 1*x2 <= 46',
        '5*x0 + 4*x1 <= 44',
        '5*x0 + 4*x1 + 1*x2 <= 44',
        '9*x0 + 9*x2 <= 61',
        '16*x1 + 9*x2 <= 37',
        '9*x0^2 + 16*x1^2 <= 85',
        '9*x0 + 16*x1 + 9*x2 <= 85'
    ]
}
```

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

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="x0", lb=0, vtype=gurobi.GRB.CONTINUOUS)  # hours worked by Bill
    x1 = model.addVar(name="x1", lb=0, vtype=gurobi.GRB.INTEGER)  # hours worked by Ringo
    x2 = model.addVar(name="x2", lb=0, vtype=gurobi.GRB.INTEGER)  # hours worked by Mary

    # Define objective function
    model.setObjective(5.86*x0**2 + 3.14*x0*x1 + 9.5*x1*x2 + 9.69*x2**2 + 6.42*x0 + 9.76*x1, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(9*x0**2 + 9*x2**2 >= 29)
    model.addConstr(9*x0**2 + 16*x1**2 >= 21)
    model.addConstr(-8*x1**2 + 4*x2**2 >= 0)
    model.addConstr(5*x0 + 1*x2 <= 97)
    model.addConstr(4*x1 + 1*x2 <= 46)
    model.addConstr(5*x0 + 4*x1 <= 44)
    model.addConstr(5*x0 + 4*x1 + 1*x2 <= 44)
    model.addConstr(9*x0 + 9*x2 <= 61)
    model.addConstr(16*x1 + 9*x2 <= 37)
    model.addConstr(9*x0**2 + 16*x1**2 <= 85)
    model.addConstr(9*x0 + 16*x1 + 9*x2 <= 85)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("x0: ", x0.varValue)
        print("x1: ", x1.varValue)
        print("x2: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```