## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank', 'hours worked by Peggy', and 'hours worked by Bill', which can be represented symbolically as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $7x_1^2 + 4x_1x_2 + 9x_2^2 + 8x_2x_3 + 9x_3^2 + 2x_1 + 2x_2 + 7x_3$.

## 3: Define the constraints in symbolic notation
The constraints are:
- $0.25x_1 \leq 235$
- $1.22x_2 \leq 235$
- $0.39x_3 \leq 235$
- $1.22x_2 + 0.39x_3 \geq 71$
- $0.25x_1 + 1.22x_2 \geq 46$
- $0.25x_1 + 1.22x_2 + 0.39x_3 \geq 51$
- $1.22x_2 + 0.39x_3 \leq 199$
- $0.25x_1 + 1.22x_2 \leq 176$
- $0.25x_1 + 0.39x_3 \leq 98$
- $0.25x_1 + 1.22x_2 + 0.39x_3 \leq 98$
- $x_3$ is an integer.

## 4: Create a symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Peggy'), ('x3', 'hours worked by Bill')],
'objective_function': '7*x1^2 + 4*x1*x2 + 9*x2^2 + 8*x2*x3 + 9*x3^2 + 2*x1 + 2*x2 + 7*x3',
'constraints': [
'0.25*x1 <= 235',
'1.22*x2 <= 235',
'0.39*x3 <= 235',
'1.22*x2 + 0.39*x3 >= 71',
'0.25*x1 + 1.22*x2 >= 46',
'0.25*x1 + 1.22*x2 + 0.39*x3 >= 51',
'1.22*x2 + 0.39*x3 <= 199',
'0.25*x1 + 1.22*x2 <= 176',
'0.25*x1 + 0.39*x3 <= 98',
'0.25*x1 + 1.22*x2 + 0.39*x3 <= 98',
'x3 % 1 == 0'  # Integer constraint for x3
]
}
```

## Step 5: Implement the optimization problem using Gurobi
```python
import gurobi as gp
from gurobi import GRB

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_Hank", lb=-GRB.INFINITY, ub=GRB.INFINITY)
x2 = m.addVar(name="hours_worked_by_Peggy", lb=-GRB.INFINITY, ub=GRB.INFINITY)
x3 = m.addVar(name="hours_worked_by_Bill", lb=-GRB.INFINITY, ub=GRB.INFINITY, vtype=GRB.INTEGER)

# Objective function
m.setObjective(7*x1**2 + 4*x1*x2 + 9*x2**2 + 8*x2*x3 + 9*x3**2 + 2*x1 + 2*x2 + 7*x3, GRB.MAXIMIZE)

# Constraints
m.addConstr(0.25*x1 <= 235)
m.addConstr(1.22*x2 <= 235)
m.addConstr(0.39*x3 <= 235)
m.addConstr(1.22*x2 + 0.39*x3 >= 71)
m.addConstr(0.25*x1 + 1.22*x2 >= 46)
m.addConstr(0.25*x1 + 1.22*x2 + 0.39*x3 >= 51)
m.addConstr(1.22*x2 + 0.39*x3 <= 199)
m.addConstr(0.25*x1 + 1.22*x2 <= 176)
m.addConstr(0.25*x1 + 0.39*x3 <= 98)
m.addConstr(0.25*x1 + 1.22*x2 + 0.39*x3 <= 98)

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Hank: {x1.varValue}")
    print(f"Hours worked by Peggy: {x2.varValue}")
    print(f"Hours worked by Bill: {x3.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```