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

## Step 2: Convert the given resources/attributes into a usable form
The resources/attributes given are:
- $r_0$: work quality rating with upper bound 84, $x_0 = 12$, $x_1 = 20$
- $r_1$: productivity rating with upper bound 92, $x_0 = 2$, $x_1 = 12$

## 3: Formulate the objective function
The objective function to minimize is: $1 \cdot x_1^2 + 6 \cdot x_1 \cdot x_2 + 2 \cdot x_2^2 + 6 \cdot x_1 + 8 \cdot x_2$

## 4: Define the constraints
1. $12x_1 \geq 12$ (Mary's work quality rating)
2. $2x_1 \geq 2$ (Mary's productivity rating)
3. $20x_2 \geq 20$ (George's work quality rating)
4. $12x_2 \geq 12$ (George's productivity rating)
5. $12^2x_1^2 + 20^2x_2^2 \geq 38$ (combined work quality rating squared)
6. $12x_1 + 20x_2 \geq 38$ (combined work quality rating)
7. $2^2x_1^2 + 12^2x_2^2 \geq 28$ (combined productivity rating squared)
8. $2x_1 + 12x_2 \geq 28$ (combined productivity rating)
9. $-10x_1 + 7x_2 \geq 0$
10. $12^2x_1^2 + 20^2x_2^2 \leq 55$ (upper bound on combined work quality rating squared)
11. $2x_1 + 12x_2 \leq 52$ (upper bound on combined productivity rating)

## 5: Simplify constraints for clarity
1. $x_1 \geq 1$
2. $x_1 \geq 1$
3. $x_2 \geq 1$
4. $x_2 \geq 1$
5. $144x_1^2 + 400x_2^2 \geq 38$
6. $12x_1 + 20x_2 \geq 38$
7. $4x_1^2 + 144x_2^2 \geq 28$
8. $2x_1 + 12x_2 \geq 28$
9. $-10x_1 + 7x_2 \geq 0$
10. $144x_1^2 + 400x_2^2 \leq 55$
11. $2x_1 + 12x_2 \leq 52$

## 6: Create the symbolic representation
$\textbf{sym\_variables:} [('x1', 'hours worked by Mary'), ('x2', 'hours worked by George')]$

$\textbf{objective\_function:} 1x_1^2 + 6x_1x_2 + 2x_2^2 + 6x_1 + 8x_2$

$\textbf{constraints:}$
- $x_1 \geq 1$
- $x_1 \geq 1$
- $x_2 \geq 1$
- $x_2 \geq 1$
- $144x_1^2 + 400x_2^2 \geq 38$
- $12x_1 + 20x_2 \geq 38$
- $4x_1^2 + 144x_2^2 \geq 28$
- $2x_1 + 12x_2 \geq 28$
- $-10x_1 + 7x_2 \geq 0$
- $144x_1^2 + 400x_2^2 \leq 55$
- $2x_1 + 12x_2 \leq 52$

## 7: Write the Gurobi code
```python
import gurobi

def optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="hours_worked_by_Mary")
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="hours_worked_by_George")
    
    # Objective function
    model.setObjective(1*x1**2 + 6*x1*x2 + 2*x2**2 + 6*x1 + 8*x2, gurobi.GRB.MINIMIZE)
    
    # Constraints
    model.addConstr(x1 >= 1, name="Mary_work_quality")
    model.addConstr(x1 >= 1, name="Mary_productivity")
    model.addConstr(x2 >= 1, name="George_work_quality")
    model.addConstr(x2 >= 1, name="George_productivity")
    model.addConstr(144*x1**2 + 400*x2**2 >= 38, name="combined_work_quality_squared")
    model.addConstr(12*x1 + 20*x2 >= 38, name="combined_work_quality")
    model.addConstr(4*x1**2 + 144*x2**2 >= 28, name="combined_productivity_squared")
    model.addConstr(2*x1 + 12*x2 >= 28, name="combined_productivity")
    model.addConstr(-10*x1 + 7*x2 >= 0, name="hour_constraint")
    model.addConstr(144*x1**2 + 400*x2**2 <= 55, name="upper_bound_work_quality_squared")
    model.addConstr(2*x1 + 12*x2 <= 52, name="upper_bound_productivity")
    
    # Optimize
    model.optimize()
    
    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Mary: ", x1.varValue)
        print("Hours worked by George: ", x2.varValue)
    else:
        print("No solution found")

optimization_problem()
```

```json
{
    'sym_variables': [('x1', 'hours worked by Mary'), ('x2', 'hours worked by George')],
    'objective_function': '1*x1^2 + 6*x1*x2 + 2*x2^2 + 6*x1 + 8*x2',
    'constraints': [
        'x1 >= 1',
        'x1 >= 1',
        'x2 >= 1',
        'x2 >= 1',
        '144*x1^2 + 400*x2^2 >= 38',
        '12*x1 + 20*x2 >= 38',
        '4*x1^2 + 144*x2^2 >= 28',
        '2*x1 + 12*x2 >= 28',
        '-10*x1 + 7*x2 >= 0',
        '144*x1^2 + 400*x2^2 <= 55',
        '2*x1 + 12*x2 <= 52'
    ]
}
```