## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic representation of the problem, including the variables, objective function, and constraints.

## Step 2: Identify the variables and their corresponding symbolic notation
The variables are:
- hours worked by Peggy: $x_0$
- hours worked by Bill: $x_1$
- hours worked by Paul: $x_2$

## Step 3: Define the objective function in symbolic notation
The objective function to minimize is: $6x_0^2 + 9x_0x_1 + 2x_2^2 + x_0$

## 4: List the constraints in symbolic notation
The constraints are:
- $x_0 \geq 0$
- $x_1 \geq 0$
- $x_2 \geq 0$
- $x_0^2 + x_1^2 \geq 29$
- $x_0 + x_2 \geq 23$
- $x_0 + x_1 + x_2 \geq 23$
- $8x_0 + 16x_2 \geq 25$
- $8x_0 + 2x_1 + 16x_2 \geq 25$
- $5x_0^2 - 4x_2^2 \geq 0$
- $7x_0 - 10x_1 \geq 0$
- $8x_0^2 + 2x_1^2 \leq 108$
- $2x_1 + 16x_2 \leq 86$

## 5: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [('x0', 'hours worked by Peggy'), ('x1', 'hours worked by Bill'), ('x2', 'hours worked by Paul')],
    'objective_function': '6*x0^2 + 9*x0*x1 + 2*x2^2 + x0',
    'constraints': [
        'x0^2 + x1^2 >= 29',
        'x0 + x2 >= 23',
        'x0 + x1 + x2 >= 23',
        '8*x0 + 16*x2 >= 25',
        '8*x0 + 2*x1 + 16*x2 >= 25',
        '5*x0^2 - 4*x2^2 >= 0',
        '7*x0 - 10*x1 >= 0',
        '8*x0^2 + 2*x1^2 <= 108',
        '2*x1 + 16*x2 <= 86'
    ]
}
```

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

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Peggy
x1 = m.addVar(name="x1", lb=0)  # hours worked by Bill
x2 = m.addVar(name="x2", lb=0)  # hours worked by Paul

# Define the objective function
m.setObjective(6 * x0**2 + 9 * x0 * x1 + 2 * x2**2 + x0, gp.GRB.MINIMIZE)

# Define the constraints
m.addConstr(x0**2 + x1**2 >= 29)
m.addConstr(x0 + x2 >= 23)
m.addConstr(x0 + x1 + x2 >= 23)
m.addConstr(8 * x0 + 16 * x2 >= 25)
m.addConstr(8 * x0 + 2 * x1 + 16 * x2 >= 25)
m.addConstr(5 * x0**2 - 4 * x2**2 >= 0)
m.addConstr(7 * x0 - 10 * x1 >= 0)
m.addConstr(8 * x0**2 + 2 * x1**2 <= 108)
m.addConstr(2 * x1 + 16 * x2 <= 86)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Peggy: {x0.varValue}")
    print(f"Hours worked by Bill: {x1.varValue}")
    print(f"Hours worked by Paul: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```