To solve the optimization problem described, we need to maximize the objective function $5x_0 + 7x_1$ where $x_0$ represents the hours worked by Bobby and $x_1$ represents the hours worked by Bill. The constraints given are:

1. $14.16x_0 + 5.77x_1 \geq 8$
2. $-4x_0 + 5x_1 \geq 0$
3. $14.16x_0 + 5.77x_1 \leq 39$
4. Since the total combined work quality rating should be 39 at maximum, we can consider this as an equality for optimization purposes: $14.16x_0 + 5.77x_1 = 39$
5. $x_0$ must be an integer (hours worked by Bobby).
6. $x_1$ must be an integer (hours worked by Bill).

Given the constraints, especially with the work quality ratings and their bounds, we aim to find integer values of $x_0$ and $x_1$ that satisfy all conditions while maximizing the objective function.

Here's how we can represent this problem using Gurobi in Python:

```python
from gurobipy import *

# Create a model
m = Model("Work Hours Optimization")

# Define variables
x0 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Bobby")
x1 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Bill")

# Set the objective function
m.setObjective(5*x0 + 7*x1, GRB.MAXIMIZE)

# Add constraints
m.addConstr(14.16*x0 + 5.77*x1 >= 8, "work_quality_min")
m.addConstr(-4*x0 + 5*x1 >= 0, "balance_hours")
m.addConstr(14.16*x0 + 5.77*x1 <= 39, "work_quality_max")
# Considering the equality for optimization purposes
m.addConstr(14.16*x0 + 5.77*x1 == 39, "work_quality_exact")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Bobby: {x0.x}")
    print(f"Hours worked by Bill: {x1.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```