## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill' and 'hours worked by Peggy', which we can denote as $x_0$ and $x_1$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $8.39x_0^2 + 8.27x_0x_1 + 9.56x_1^2 + 8.29x_0 + 9.12x_1$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $15x_0 \leq 88$ (not directly given but implied by $x_0$ having an upper bound related to 'dollar cost per hour' and its value for Bill)
- $21x_0 \leq 106$ (not directly given but implied by $x_0$ having an upper bound related to 'paperwork competence rating' and its value for Bill)
- $4x_0 \leq 118$ (not directly given but implied by $x_0$ having an upper bound related to 'productivity rating' and its value for Bill)
- $2x_1 \leq 88$ (similar reasoning for Peggy)
- $9x_1 \leq 106$ 
- $26x_1 \leq 118$
- $15x_0 + 2x_1 \geq 43$
- $21x_0^2 + 9x_1^2 \geq 40$
- $4x_0^2 + 26x_1^2 \geq 27$
- $9x_0^2 - 6x_1^2 \geq 0$
- $15^2x_0^2 + 2^2x_1^2 \leq 67$ or more accurately $225x_0^2 + 4x_1^2 \leq 67$ (assuming 'dollar cost per hour' squared and summed)
- $21^2x_0^2 + 9^2x_1^2 \leq 99$ or more accurately $441x_0^2 + 81x_1^2 \leq 99$
- $4x_0 + 26x_1 \leq 106$

## 4: Correct and Clarify Constraints
Correcting and clarifying:
- The actual constraints from the problem statement should directly relate to the variables $x_0$ and $x_1$ without misinterpretation.
- The constraints should be:
  - $x_0 \geq 0$, $x_1 \geq 0$ (implicit non-negativity)
  - $15x_0 + 2x_1 \geq 43$
  - $21x_0 + 9x_1 \geq 40$
  - $4x_0 + 26x_1 \geq 27$
  - $9x_0^2 - 6x_1^2 \geq 0$
  - $225x_0^2 + 4x_1^2 \leq 67$
  - $441x_0^2 + 81x_1^2 \leq 99$
  - $4x_0 + 26x_1 \leq 106$
  - $x_1$ is an integer

## 5: Symbolic Representation
The symbolic representation is:
```json
{
  'sym_variables': [('x0', 'hours worked by Bill'), ('x1', 'hours worked by Peggy')],
  'objective_function': '8.39*x0^2 + 8.27*x0*x1 + 9.56*x1^2 + 8.29*x0 + 9.12*x1',
  'constraints': [
    '15*x0 + 2*x1 >= 43',
    '21*x0^2 + 9*x1^2 >= 40',
    '4*x0^2 + 26*x1^2 >= 27',
    '9*x0^2 - 6*x1^2 >= 0',
    '225*x0^2 + 4*x1^2 <= 67',
    '441*x0^2 + 81*x1^2 <= 99',
    '4*x0 + 26*x1 <= 106',
    'x1 % 1 == 0'  # x1 is an integer
  ]
}
```

## 6: Gurobi Code
Now, let's write the Gurobi code for this problem:
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x0 = model.addVar(lb=0, name="hours_worked_by_Bill")  # No upper bound given, assuming continuous
    x1 = model.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="hours_worked_by_Peggy")
    
    # Objective function
    model.setObjective(8.39*x0**2 + 8.27*x0*x1 + 9.56*x1**2 + 8.29*x0 + 9.12*x1, gurobi.GRB.MINIMIZE)
    
    # Constraints
    model.addConstr(15*x0 + 2*x1 >= 43)
    model.addConstr(21*x0**2 + 9*x1**2 >= 40)
    model.addConstr(4*x0**2 + 26*x1**2 >= 27)
    model.addConstr(9*x0**2 - 6*x1**2 >= 0)
    model.addConstr(225*x0**2 + 4*x1**2 <= 67)
    model.addConstr(441*x0**2 + 81*x1**2 <= 99)
    model.addConstr(4*x0 + 26*x1 <= 106)
    
    # Optimize
    model.optimize()
    
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Bill: {x0.varValue}")
        print(f"Hours worked by Peggy: {x1.varValue}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has a non-optimal status.")

solve_optimization_problem()
```