To create a Gurobi model that represents the given problem, we'll need to define variables for the hours worked by each person (Peggy, George, Bobby, Dale, and Hank), set up the objective function based on the provided equation, and then add all the constraints as inequalities or equalities.

Let's denote:
- $x_P$ as the number of hours worked by Peggy,
- $x_G$ as the number of hours worked by George,
- $x_B$ as the number of hours worked by Bobby,
- $x_D$ as the number of hours worked by Dale,
- $x_H$ as the number of hours worked by Hank.

The objective function, based on the given equation (though it seems there might be a typo or missing information regarding the exact objective), appears to involve terms related to hours worked and possibly costs or productivity. However, since the specific objective is not clearly defined in your problem statement, we will assume a placeholder objective for demonstration purposes.

Given constraints:
- Integer constraints for some variables,
- Bounds on total dollar cost per hour for various combinations of workers,
- Bounds on work quality ratings for different worker combinations,
- Bounds on likelihood to quit index for various worker groups,
- Productivity bounds for different combinations of workers.

Here's a simplified representation of how you might set up the model in Gurobi, focusing on key aspects:

```python
from gurobipy import *

# Create a new model
m = Model("Work Hours Optimization")

# Define variables (hours worked by each person)
x_P = m.addVar(name="Peggy", lb=0)  # Non-negative, can be fractional
x_G = m.addVar(name="George", vtype=GRB.INTEGER, lb=0)  # Integer, non-negative
x_B = m.addVar(name="Bobby", lb=0)  # Can be fractional
x_D = m.addVar(name="Dale", lb=0)  # Can be fractional
x_H = m.addVar(name="Hank", vtype=GRB.INTEGER, lb=0)  # Integer, non-negative

# Placeholder objective function (replace with actual objective)
m.setObjective(9*x_P + x_G - 2*x_B + 3*x_D + x_H, GRB.MINIMIZE)

# Constraints
# Example constraints based on provided inequalities
m.addConstr(x_P + x_G <= 118, "Total cost constraint")
m.addConstr(x_G**2 + x_B**2 <= 153, "Cost constraint squared terms")
m.addConstr(-2*x_P + 9*x_D >= 0, "Peggy and Dale productivity")

# Adding more constraints as needed based on the problem description
# This would include all other provided inequalities regarding work quality,
# likelihood to quit, productivity, etc.

m.optimize()

if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    for v in m.getVars():
        print(f"{v.varName}: {v.x}")
else:
    print("No optimal solution found.")
```

This code sets up a basic model with variables representing the hours worked by each person, adds some example constraints (you would need to add all relevant constraints from your problem description), and then solves the optimization problem. Note that you'll need to replace the placeholder objective function with the actual one based on your specific needs.

Remember, this is a simplified representation. The actual implementation might require additional steps or adjustments based on the full specifics of your problem, including correctly formulating all constraints and possibly handling more complex relationships between variables.