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

## Step 2: Convert the natural language objects into symbolic notation
The symbolic representation of the variables and their corresponding natural language objects are:
- $x_0$ : 'hours worked by Peggy'
- $x_1$ : 'hours worked by Hank'
- $x_2$ : 'hours worked by George'

## 3: Define the objective function in symbolic notation
The objective function to minimize is $8x_0x_1 + x_2^2 + x_0$.

## 4: Define the constraints in symbolic notation
The constraints are:
- $x_0 \geq 0$ (Implicit, as hours cannot be negative, but not explicitly stated)
- $x_1 \geq 0$ (Implicit, as hours cannot be negative, but not explicitly stated)
- $x_2 \geq 0$ (Implicit, as hours cannot be negative, but not explicitly stated)
- $x_0^2 \cdot 1 + x_1^2 \cdot 8 + x_2^2 \cdot 6 \leq 108$ (Paperwork competence rating constraint, but it seems there was a misunderstanding in direct translation. The correct translation based on given attributes is not directly provided, so we focus on given constraints)
- $x_0 \cdot 1 + x_1 \cdot 2 + x_2 \cdot 3 \leq 108$ (Work quality rating constraint, similarly)
- $x_1^2 + x_2^2 \geq 28$ (Minimum combined paperwork competence rating)
- $x_0 + x_1 + x_2 \geq 28$ (Minimum combined paperwork competence rating)
- $x_1 + x_2 \geq 23$ (Minimum combined work quality rating from Hank and George)
- $x_0^2 + x_2^2 \geq 24$ (Work quality rating constraint)
- $x_0^2 + x_1^2 + x_2^2 \geq 21$ (Work quality rating constraint)
- $x_0 + x_1 + x_2 \geq 21$ (Work quality rating constraint)
- $9x_0^2 - 7x_1^2 \geq 0$ (Constraint on hours worked by Peggy and Hank)
- $x_0 + x_2 \leq 79$ (Constraint on work quality rating from Peggy and George)
- $x_1^2 + x_2^2 \leq 58$ (Constraint on work quality rating from Hank and George)

## 5: Provide the symbolic representation of the problem
```json
{
'sym_variables': [('x0', 'hours worked by Peggy'), ('x1', 'hours worked by Hank'), ('x2', 'hours worked by George')],
'objective_function': '8*x0*x1 + x2^2 + x0',
'constraints': [
    'x1^2 + x2^2 >= 28',
    'x0 + x1 + x2 >= 28',
    'x1 + x2 >= 23',
    'x0^2 + x2^2 >= 24',
    'x0^2 + x1^2 + x2^2 >= 21',
    'x0 + x1 + x2 >= 21',
    '9*x0^2 - 7*x1^2 >= 0',
    'x0 + x2 <= 79',
    'x1^2 + x2^2 <= 58',
    'x0 >= 0',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 6: Implement the optimization problem using Gurobi
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Peggy
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Hank
    x2 = model.addVar(name="x2", lb=0)  # hours worked by George

    # Objective function
    model.setObjective(8 * x0 * x1 + x2 ** 2 + x0, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x1 ** 2 + x2 ** 2 >= 28)
    model.addConstr(x0 + x1 + x2 >= 28)
    model.addConstr(x1 + x2 >= 23)
    model.addConstr(x0 ** 2 + x2 ** 2 >= 24)
    model.addConstr(x0 ** 2 + x1 ** 2 + x2 ** 2 >= 21)
    model.addConstr(x0 + x1 + x2 >= 21)
    model.addConstr(9 * x0 ** 2 - 7 * x1 ** 2 >= 0)
    model.addConstr(x0 + x2 <= 79)
    model.addConstr(x1 ** 2 + x2 ** 2 <= 58)

    # Optimize
    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Peggy: {x0.varValue}")
        print(f"Hours worked by Hank: {x1.varValue}")
        print(f"Hours worked by George: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    elif model.status == gurobi.GRB.Status.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has a non-optimal status.")

optimize_problem()
```