## Step 1: Define the symbolic variables and their corresponding natural language objects.
The variables are 'hours worked by Hank', 'hours worked by Mary', 'hours worked by Ringo', 'hours worked by John', which can be denoted as $x_0$, $x_1$, $x_2$, $x_3$ respectively.

## Step 2: Translate the objective function into a symbolic representation using the defined variables.
The objective function to maximize is $8x_0^2 + 5x_0x_2 + 7x_0x_3 + 5x_1^2 + 4x_2^2 + 4x_3^2 + 3x_0 + 6x_3$.

## 3: List the constraints in symbolic notation.
1. $16x_0 \leq 432$ (Hank's computer competence rating is 16, but this seems to be an upper bound constraint, not a direct limit on $x_0$. Assuming $x_0$ is directly limited by its coefficient in $r_0$: $16x_0$),
2. $13x_0 \leq 335$ (Hank's productivity rating),
3. $20x_1 \leq 432$ (Mary's computer competence rating),
4. $4x_1 \leq 335$ (Mary's productivity rating),
5. $16x_2 \leq 432$ (Ringo's computer competence rating),
6. $12x_2 \leq 335$ (Ringo's productivity rating),
7. $x_3 \leq 432$ (John's computer competence rating, assuming),
8. $x_3 \leq 335$ (John's productivity rating, assuming),
9. $13x_0 + 12x_2 + x_3 \geq 77$ (Total combined productivity rating),
10. $20x_1 + x_3 \leq 156$ (Total combined computer competence rating from Mary and John),
11. $16x_0^2 + 20x_1^2 \leq 326$ (Total combined computer competence rating from Hank and Mary squared),
12. $16x_0 + 16x_2 \leq 404$ (Total combined computer competence rating from Hank and Ringo),
13. $16x_0^2 + x_3^2 \leq 213$ (Total combined computer competence rating from Hank squared and John squared),
14. $16x_2 + x_3 \leq 146$ (Total combined computer competence rating from Ringo and John),
15. $16x_0^2 + 20x_1^2 + x_3^2 \leq 353$ (Total combined computer competence rating from Hank, Mary, and John squared),
16. $16x_0 + 20x_1 + 16x_2 + x_3 \leq 353$ (Total combined computer competence rating),
17. $4x_1 + 12x_2 \leq 314$ (Total combined productivity rating from Mary and Ringo),
18. $4x_1 + x_3 \leq 119$ (Total combined productivity rating from Mary and John),
19. $13x_0^2 + x_3^2 \leq 97$ (Total combined productivity rating from Hank squared and John squared),
20. $13x_0^2 + 12x_2^2 \leq 218$ (Total combined productivity rating from Hank squared and Ringo squared),
21. $13x_0 + 4x_1 \leq 301$ (Total combined productivity rating from Hank and Mary),
22. $13x_0 + 4x_1 + 12x_2 + x_3 \leq 301$ (Total combined productivity rating).

## 4: Consider variable bounds.
- $x_0$ can be non-whole,
- $x_1$ can be non-fractional (implying it can be any real number),
- $x_2$ and $x_3$ must be non-fractional (integers).

## 5: Formulate the problem in Gurobi Python.
```python
import gurobi

# Define the model
model = gurobi.Model()

# Define variables
x0 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x0")  # hours worked by Hank
x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x1")  # hours worked by Mary
x2 = model.addVar(lb=0, ub=gurobi.GRB.INFINITY, name="x2", vtype=gurobi.GRB.INTEGER)  # hours worked by Ringo
x3 = model.addVar(lb=0, ub=gurobi.GRB.INFINITY, name="x3", vtype=gurobi.GRB.INTEGER)  # hours worked by John

# Objective function
model.setObjective(8*x0**2 + 5*x0*x2 + 7*x0*x3 + 5*x1**2 + 4*x2**2 + 4*x3**2 + 3*x0 + 6*x3, gurobi.GRB.MAXIMIZE)

# Constraints
model.addConstr(16*x0 <= 432)
model.addConstr(13*x0 <= 335)
model.addConstr(20*x1 <= 432)
model.addConstr(4*x1 <= 335)
model.addConstr(16*x2 <= 432)
model.addConstr(12*x2 <= 335)
model.addConstr(x3 <= 432)
model.addConstr(x3 <= 335)
model.addConstr(13*x0 + 12*x2 + x3 >= 77)
model.addConstr(20*x1 + x3 <= 156)
model.addConstr(16*x0**2 + 20*x1**2 <= 326)
model.addConstr(16*x0 + 16*x2 <= 404)
model.addConstr(16*x0**2 + x3**2 <= 213)
model.addConstr(16*x2 + x3 <= 146)
model.addConstr(16*x0**2 + 20*x1**2 + x3**2 <= 353)
model.addConstr(16*x0 + 20*x1 + 16*x2 + x3 <= 353)
model.addConstr(4*x1 + 12*x2 <= 314)
model.addConstr(4*x1 + x3 <= 119)
model.addConstr(13*x0**2 + x3**2 <= 97)
model.addConstr(13*x0**2 + 12*x2**2 <= 218)
model.addConstr(13*x0 + 4*x1 <= 301)
model.addConstr(13*x0 + 4*x1 + 12*x2 + x3 <= 301)

# 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)
    print("x3: ", x3.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic Representation
```json
{
    'sym_variables': [('x0', 'hours worked by Hank'), ('x1', 'hours worked by Mary'), ('x2', 'hours worked by Ringo'), ('x3', 'hours worked by John')],
    'objective_function': '8*x0^2 + 5*x0*x2 + 7*x0*x3 + 5*x1^2 + 4*x2^2 + 4*x3^2 + 3*x0 + 6*x3',
    'constraints': [
        '16*x0 <= 432',
        '13*x0 <= 335',
        '20*x1 <= 432',
        '4*x1 <= 335',
        '16*x2 <= 432',
        '12*x2 <= 335',
        'x3 <= 432',
        'x3 <= 335',
        '13*x0 + 12*x2 + x3 >= 77',
        '20*x1 + x3 <= 156',
        '16*x0^2 + 20*x1^2 <= 326',
        '16*x0 + 16*x2 <= 404',
        '16*x0^2 + x3^2 <= 213',
        '16*x2 + x3 <= 146',
        '16*x0^2 + 20*x1^2 + x3^2 <= 353',
        '16*x0 + 20*x1 + 16*x2 + x3 <= 353',
        '4*x1 + 12*x2 <= 314',
        '4*x1 + x3 <= 119',
        '13*x0^2 + x3^2 <= 97',
        '13*x0^2 + 12*x2^2 <= 218',
        '13*x0 + 4*x1 <= 301',
        '13*x0 + 4*x1 + 12*x2 + x3 <= 301'
    ]
}
```