To tackle this problem, we first need to understand and translate the given natural language description into a symbolic representation of an optimization problem. This involves defining variables, formulating the objective function, and listing all constraints.

Given variables:
- Hours worked by Hank
- Hours worked by Mary
- Hours worked by Paul

These can be represented symbolically as \(x_1\), \(x_2\), and \(x_3\) respectively.

Objective Function:
The goal is to minimize \(1.4 \times x_1 \times x_2 + 3.28 \times x_1 \times x_3 + 4.19 \times x_2^2 + 4.96 \times x_1 + 3.96 \times x_2\).

Constraints:
1. \(x_1\)'s likelihood to quit index is 3.
2. \(x_2\)'s likelihood to quit index is 7.
3. \(x_3\)'s likelihood to quit index is 7.
4. The total combined likelihood to quit index from \(x_1\) and \(x_3\) should be at least 4.
5. The total combined likelihood to quit index from \(x_2\) and \(x_3\) should be at least 4.
6. The total combined likelihood to quit index from \(x_1\), \(x_2\), and \(x_3\) should be greater than or equal to 4.
7. \(5 \times x_1 - 8 \times x_2 \geq 0\).

Given the problem statement, it seems there might be some confusion in directly translating "likelihood to quit index" into constraints without specific relationships provided between hours worked and these indices. However, based on the given information:

- Let's assume \(x_1\), \(x_2\), and \(x_3\) represent the hours worked by Hank, Mary, and Paul respectively.
- The "likelihood to quit index" for each seems to be a constant attribute rather than a function of their hours worked. Thus, constraints 1, 2, and 3 might not directly influence the optimization problem as they are more descriptive attributes of Hank, Mary, and Paul rather than constraints on \(x_1\), \(x_2\), and \(x_3\).

However, to follow through with the symbolic representation requested:

```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': [
    'x1 >= 0', 
    'x2 >= 0', 
    'x3 >= 0', 
    '5*x1 - 8*x2 >= 0'
  ]
}
```

Note: The constraints related to "likelihood to quit index" are not directly incorporated as they seem to represent attributes rather than functional constraints on the variables. Also, since all hours worked can be fractional, we don't need integer constraints.

For the Gurobi code:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x1 = m.addVar(lb=0, name="hours_worked_by_Hank")
x2 = m.addVar(lb=0, name="hours_worked_by_Mary")
x3 = m.addVar(lb=0, name="hours_worked_by_Paul")

# Set the objective function
m.setObjective(1.4*x1*x2 + 3.28*x1*x3 + 4.19*x2**2 + 4.96*x1 + 3.96*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(5*x1 - 8*x2 >= 0, name="constraint_1")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print("Hours worked by Hank:", x1.x)
    print("Hours worked by Mary:", x2.x)
    print("Hours worked by Paul:", x3.x)
else:
    print("No optimal solution found")
```