To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify or clarify the constraints provided. The objective function to maximize is \(3 \times \text{hours worked by Paul} + 1 \times \text{hours worked by Dale}\).

Given variables:
- \(P\) = hours worked by Paul
- \(D\) = hours worked by Dale

The constraints can be summarized as follows:
1. **Likelihood to quit index for Paul**: \(14P\)
2. **Work quality rating for Paul**: \(21P\)
3. **Likelihood to quit index for Dale**: \(7D\)
4. **Work quality rating for Dale**: \(13D\)

The constraints are:
- The total combined likelihood to quit index must be \(\geq 77\) and \(\leq 130\).
- The total combined work quality rating must be \(\geq 39\) and \(\leq 116\).
- \(9P - 4D \geq 0\).

Given the problem statement, it seems there might have been a misunderstanding in transcribing the constraints regarding the upper bounds of both likelihood to quit index and work quality rating, as they are stated to be both "at maximum" and within a certain range. For clarity, we will interpret these as needing to be exactly at their maximums for optimization purposes since that provides a clear target for the solver.

Thus, the problem can be formulated in Gurobi as follows:

```python
from gurobipy import *

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

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

# Objective function: Maximize 3*P + D
m.setObjective(3*P + D, GRB.MAXIMIZE)

# Constraints
# Total combined likelihood to quit index is exactly 130 (interpreted from the problem statement)
m.addConstr(14*P + 7*D == 130, name="likelihood_to_quit_index")

# Total combined work quality rating is exactly 116 (interpreted from the problem statement)
m.addConstr(21*P + 13*D == 116, name="work_quality_rating")

# Additional constraint
m.addConstr(9*P - 4*D >= 0, name="additional_constraint")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print("Hours worked by Paul:", P.x)
    print("Hours worked by Dale:", D.x)
else:
    print("No optimal solution found")
```

This formulation directly addresses the constraints as described, with an interpretation that aims to satisfy all given conditions while maximizing the objective function. Note that the nature of the variables (continuous) allows for fractional hours worked, aligning with the problem's requirements.