## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bobby' and 'hours worked by Bill', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $5x_1 + 7x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
1. $14.16x_1 + 5.77x_2 \geq 8$
2. $-4x_1 + 5x_2 \geq 0$
3. $14.16x_1 + 5.77x_2 \leq 39$
4. $x_1$ must be an integer
5. $x_2$ must be an integer

## 4: Convert the problem into a Gurobi-compatible model
We will use Gurobi's Python API to model this problem.

## 5: Write the Gurobi code
```python
import gurobi as gp

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_Bobby", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="hours_worked_by_Bill", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(5 * x1 + 7 * x2, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(14.16 * x1 + 5.77 * x2 >= 8, name="work_quality_rating_constraint1")
m.addConstr(-4 * x1 + 5 * x2 >= 0, name="work_quality_rating_constraint2")
m.addConstr(14.16 * x1 + 5.77 * x2 <= 39, name="work_quality_rating_constraint3")

# Optimize the model
m.optimize()

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

## 6: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Bobby'), ('x2', 'hours worked by Bill')],
    'objective_function': '5*x1 + 7*x2',
    'constraints': [
        '14.16*x1 + 5.77*x2 >= 8',
        '-4*x1 + 5*x2 >= 0',
        '14.16*x1 + 5.77*x2 <= 39',
        'x1 is an integer',
        'x2 is an integer'
    ]
}
```