## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill', 'hours worked by Dale', and 'hours worked by Mary', 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:
- $r_0$: productivity rating with $x_0 = 5$, $x_1 = 7$, $x_2 = 7$ and upper bound $92$.
- $r_1$: work quality rating with $x_0 = 1$, $x_1 = 6$, $x_2 = 3$ and upper bound $34$.

## 3: Formulate the objective function
The objective function to maximize is $5x_0^2 + 2x_0x_1 + 4x_0x_2 + 2x_1^2 + 3x_0 + x_1$.

## 4: List all constraints
1. $5x_0 \leq 92$ (productivity rating for Bill)
2. $1x_0 \leq 34$ (work quality rating for Bill)
3. $7x_1 \leq 92$ (productivity rating for Dale)
4. $6x_1 \leq 34$ (work quality rating for Dale)
5. $7x_2 \leq 92$ (productivity rating for Mary)
6. $3x_2 \leq 34$ (work quality rating for Mary)
7. $7x_1 + 7x_2 \geq 13$ (combined productivity rating for Dale and Mary)
8. $5x_0 + 7x_1 \geq 28$ (combined productivity rating for Bill and Dale)
9. $6x_1 + 3x_2 \geq 8$ (combined work quality rating for Dale and Mary)
10. $1x_0 + 3x_2 \geq 8$ (combined work quality rating for Bill and Mary)
11. $5^2x_0^2 + 7^2x_1^2 \leq 67$ (combined productivity rating squared for Bill and Dale)
12. $5x_0 + 7x_2 \leq 31$ (combined productivity rating for Bill and Mary)
13. $5x_0 + 7x_1 + 7x_2 \leq 31$ (combined productivity rating for all)
14. $1x_0 + 6x_1 \leq 21$ (combined work quality rating for Bill and Dale)
15. $1x_0 + 6x_1 + 3x_2 \leq 21$ (combined work quality rating for all)
16. $x_0$ is an integer (hours worked by Bill)
17. $x_1$ can be fractional (hours worked by Dale)
18. $x_2$ can be fractional (hours worked by Mary)

## 5: Symbolic Representation
```json
{
    'sym_variables': [('x0', 'hours worked by Bill'), ('x1', 'hours worked by Dale'), ('x2', 'hours worked by Mary')],
    'objective_function': '5*x0^2 + 2*x0*x1 + 4*x0*x2 + 2*x1^2 + 3*x0 + x1',
    'constraints': [
        '5*x0 <= 92',
        '1*x0 <= 34',
        '7*x1 <= 92',
        '6*x1 <= 34',
        '7*x2 <= 92',
        '3*x2 <= 34',
        '7*x1 + 7*x2 >= 13',
        '5*x0 + 7*x1 >= 28',
        '6*x1 + 3*x2 >= 8',
        '1*x0 + 3*x2 >= 8',
        '25*x0^2 + 49*x1^2 <= 67',
        '5*x0 + 7*x2 <= 31',
        '5*x0 + 7*x1 + 7*x2 <= 31',
        '1*x0 + 6*x1 <= 21',
        '1*x0 + 6*x1 + 3*x2 <= 21',
        'x0 % 1 == 0'  # integer constraint for x0
    ]
}
```

## 6: Gurobi Code
```python
import gurobi

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

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

    # Objective function
    model.setObjective(5*x0**2 + 2*x0*x1 + 4*x0*x2 + 2*x1**2 + 3*x0 + x1, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(5*x0 <= 92)
    model.addConstr(x0 <= 34)
    model.addConstr(7*x1 <= 92)
    model.addConstr(6*x1 <= 34)
    model.addConstr(7*x2 <= 92)
    model.addConstr(3*x2 <= 34)
    model.addConstr(7*x1 + 7*x2 >= 13)
    model.addConstr(5*x0 + 7*x1 >= 28)
    model.addConstr(6*x1 + 3*x2 >= 8)
    model.addConstr(x0 + 3*x2 >= 8)
    model.addConstr(25*x0**2 + 49*x1**2 <= 67)
    model.addConstr(5*x0 + 7*x2 <= 31)
    model.addConstr(5*x0 + 7*x1 + 7*x2 <= 31)
    model.addConstr(x0 + 6*x1 <= 21)
    model.addConstr(x0 + 6*x1 + 3*x2 <= 21)

    # Optimize
    model.optimize()

    # Print 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("No solution found")

solve_optimization_problem()
```