## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank', 'hours worked by Mary', and 'hours worked by Paul'. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $1.4x_1x_2 + 3.28x_1x_3 + 4.19x_2^2 + 4.96x_1 + 3.96x_2$.

## 3: Define the constraints in symbolic notation
The constraints are:
- $3x_1 \leq 39$ (Hank's likelihood to quit index is 3, and its upper bound is 39, but this seems to be an attribute rather than a direct constraint. The actual constraint from the problem statement is that Hank's index is 3, which is given)
- $7x_2 \leq 39$ (Mary's likelihood to quit index is 7)
- $7x_3 \leq 39$ (Paul's likelihood to quit index is 7)
- $3x_1 + 7x_3 \geq 4$ (total combined likelihood to quit index from Hank and Paul)
- $7x_2 + 7x_3 \geq 4$ (total combined likelihood to quit index from Mary and Paul)
- $3x_1 + 7x_2 + 7x_3 \geq 4$ (total combined likelihood to quit index from all)
- $5x_1 - 8x_2 \geq 0$ ( constraint on Hank's and Mary's hours)

## 4: Correctly interpret given attributes as constraints
Given attributes:
- {'r0': {'description': 'likelihood to quit index', 'upper_bound': 39, 'x0': 3, 'x1': 7, 'x2': 7}}
Implies direct constraints:
- $x_0 = 3$ (Hank's index is 3, but we use $x_1$ for Hank, so $3x_1 = 3$ or simply $x_1$ is free but has a value of 1 unit of index, implying $x_1$ can be scaled)
- However, it seems more accurate that $x_1=1$ (for 3 index), $x_2 = 1$ (for 7 index), $x_3 = 1$ (for 7 index) per unit of index.

## 5: Formulate the problem in Gurobi
Given the complexity and potential misinterpretation, let's focus on directly formulating the given problem.

## 6: Symbolic Representation
```json
{
    'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Mary'), ('x3', 'hours worked by Paul')],
    'objective_function': '1.4*x1*x2 + 3.28*x1*x3 + 4.19*x2^2 + 4.96*x1 + 3.96*x2',
    'constraints': [
        '3*x1 <= 39', 
        '7*x2 <= 39', 
        '7*x3 <= 39', 
        '3*x1 + 7*x3 >= 4', 
        '7*x2 + 7*x3 >= 4', 
        '3*x1 + 7*x2 + 7*x3 >= 4', 
        '5*x1 - 8*x2 >= 0'
    ]
}
```

## 7: Gurobi Code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Hank
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Mary
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Paul

    # Objective function
    model.setObjective(1.4*x1*x2 + 3.28*x1*x3 + 4.19*x2**2 + 4.96*x1 + 3.96*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(3*x1 <= 39)  
    model.addConstr(7*x2 <= 39)  
    model.addConstr(7*x3 <= 39)  
    model.addConstr(3*x1 + 7*x3 >= 4)  
    model.addConstr(7*x2 + 7*x3 >= 4)  
    model.addConstr(3*x1 + 7*x2 + 7*x3 >= 4)  
    model.addConstr(5*x1 - 8*x2 >= 0)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Hours worked by Hank: ", x1.varValue)
        print("Hours worked by Mary: ", x2.varValue)
        print("Hours worked by Paul: ", x3.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```