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

## Step 2: Convert the given attributes into symbolic constraints
Given attributes:
- 'r0': {'description': 'work quality rating', 'upper_bound': 83, 'x0': 4.71, 'x1': 1.4, 'x2': 3.16}
- 'r1': {'description': 'likelihood to quit index', 'upper_bound': 34, 'x0': 0.15, 'x1': 1.39, 'x2': 4.51}

## 3: Define the objective function in symbolic notation
The objective function to maximize is: $2x_0x_1 + 2x_0x_2 + 6x_1^2 + x_1x_2 + 6x_2^2 + 3x_0 + x_2$

## 4: List all constraints in symbolic notation
1. $4.71x_0 \leq 83$ (Implicitly, $x_0$ has no upper bound from this, but $x_0$ is Mary's work quality rating which is given as 4.71, so it's not a constraint on $x_0$ but a given)
2. $0.15x_0 \leq 34$ (Implicitly, $x_0$ has no upper bound from this)
3. $1.4x_1 \leq 83$
4. $1.39x_1 \leq 34$
5. $3.16x_2 \leq 83$
6. $4.51x_2 \leq 34$
7. $x_1^2 + x_2^2 \geq 12$
8. $1.39x_1 + 4.51x_2 \geq 6$
9. $0.15^2x_0^2 + 1.39^2x_1^2 + 4.51^2x_2^2 \geq 9$
10. $1.4x_1 + 3.16x_2 \leq 51$
11. $4.71x_0 + 1.4x_1 \leq 38$
12. $4.71x_0 + 1.4x_1 + 3.16x_2 \leq 38$
13. $0.15x_0 + 4.51x_2 \leq 27$
14. $0.15x_0 + 1.39x_1 \leq 17$
15. $1.39^2x_1^2 + 4.51^2x_2^2 \leq 17$
16. $0.15x_0 + 1.39x_1 + 4.51x_2 \leq 17$

## 5: Create a symbolic representation of the problem
```json
{
'sym_variables': [('x0', 'hours worked by Mary'), ('x1', 'hours worked by Hank'), ('x2', 'hours worked by Bill')],
'objective_function': '2*x0*x1 + 2*x0*x2 + 6*x1^2 + x1*x2 + 6*x2^2 + 3*x0 + x2',
'constraints': [
    '1.4*x1 <= 83', '1.39*x1 <= 34', '3.16*x2 <= 83', '4.51*x2 <= 34',
    'x1^2 + x2^2 >= 12', '1.39*x1 + 4.51*x2 >= 6',
    '0.0225*x0^2 + 1.9321*x1^2 + 20.4201*x2^2 >= 9',
    '1.4*x1 + 3.16*x2 <= 51', '4.71*x0 + 1.4*x1 <= 38',
    '4.71*x0 + 1.4*x1 + 3.16*x2 <= 38', '0.15*x0 + 4.51*x2 <= 27',
    '0.15*x0 + 1.39*x1 <= 17', '1.9321*x1^2 + 20.4201*x2^2 <= 17',
    '0.15*x0 + 1.39*x1 + 4.51*x2 <= 17'
]
}
```

## 6: Write the Gurobi code for the optimization problem
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="x0", lb=0, ub=None)  # hours worked by Mary
    x1 = model.addVar(name="x1", lb=0, ub=None, vtype=gurobi.GRB.INTEGER)  # hours worked by Hank
    x2 = model.addVar(name="x2", lb=0, ub=None, vtype=gurobi.GRB.INTEGER)  # hours worked by Bill

    # Objective function
    model.setObjective(2*x0*x1 + 2*x0*x2 + 6*x1**2 + x1*x2 + 6*x2**2 + 3*x0 + x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(1.4*x1 <= 83)
    model.addConstr(1.39*x1 <= 34)
    model.addConstr(3.16*x2 <= 83)
    model.addConstr(4.51*x2 <= 34)
    model.addConstr(x1**2 + x2**2 >= 12)
    model.addConstr(1.39*x1 + 4.51*x2 >= 6)
    model.addConstr(0.0225*x0**2 + 1.9321*x1**2 + 20.4201*x2**2 >= 9)
    model.addConstr(1.4*x1 + 3.16*x2 <= 51)
    model.addConstr(4.71*x0 + 1.4*x1 <= 38)
    model.addConstr(4.71*x0 + 1.4*x1 + 3.16*x2 <= 38)
    model.addConstr(0.15*x0 + 4.51*x2 <= 27)
    model.addConstr(0.15*x0 + 1.39*x1 <= 17)
    model.addConstr(1.9321*x1**2 + 20.4201*x2**2 <= 17)
    model.addConstr(0.15*x0 + 1.39*x1 + 4.51*x2 <= 17)

    # 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")

optimize_problem()
```