To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables and constraints in terms of these variables.

### Symbolic Representation

Let's denote:
- $x_1$ as the hours worked by Paul,
- $x_2$ as the hours worked by Dale.

The objective function is to maximize: $3x_1 + x_2$

Constraints are:
1. $14x_1 + 7x_2 \geq 77$ (Total combined likelihood to quit index)
2. $21x_1 + 13x_2 \geq 39$ (Total combined work quality rating)
3. $9x_1 - 4x_2 \geq 0$
4. $14x_1 + 7x_2 \leq 130$ (Upper bound on total likelihood to quit index)
5. $21x_1 + 13x_2 \leq 116$ (Upper bound on total work quality rating)

Given the specific upper bounds, constraints 4 and 5 are actually equalities in this context since they must be "at maximum," but we'll treat them as inequalities for generality.

### Symbolic Representation Output

```json
{
    'sym_variables': [('x1', 'hours worked by Paul'), ('x2', 'hours worked by Dale')],
    'objective_function': '3*x1 + x2',
    'constraints': [
        '14*x1 + 7*x2 >= 77',
        '21*x1 + 13*x2 >= 39',
        '9*x1 - 4*x2 >= 0',
        '14*x1 + 7*x2 <= 130',
        '21*x1 + 13*x2 <= 116'
    ]
}
```

### Gurobi Code

To solve this problem using Gurobi, we can implement the following Python code:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Paul")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Dale")

# Set the objective function
m.setObjective(3*x1 + x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(14*x1 + 7*x2 >= 77, "Total combined likelihood to quit index")
m.addConstr(21*x1 + 13*x2 >= 39, "Total combined work quality rating")
m.addConstr(9*x1 - 4*x2 >= 0, "Constraint on hours worked")
m.addConstr(14*x1 + 7*x2 <= 130, "Upper bound on total likelihood to quit index")
m.addConstr(21*x1 + 13*x2 <= 116, "Upper bound on total work quality rating")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Paul: {x1.x}")
    print(f"Hours worked by Dale: {x2.x}")
else:
    print("No optimal solution found")
```