To solve this optimization problem, we will use the Gurobi Python library. The objective function is to minimize a quadratic expression involving hours worked by four individuals (Hank, Dale, Jean, and George), subject to various linear and quadratic constraints related to their paperwork competence rating, organization score, work quality rating, and dollar cost per hour.

First, we need to import the necessary libraries and define the variables. We'll then set up the model, add the objective function, and include all the given constraints before solving the problem.

Here's how you can represent this problem in Gurobi Python code:

```python
from gurobipy import *

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

# Define variables (hours worked by each person)
Hank = m.addVar(lb=0, name="Hank")
Dale = m.addVar(lb=0, name="Dale")
Jean = m.addVar(lb=0, name="Jean")
George = m.addVar(lb=0, name="George")

# Objective function
m.setObjective(6.35 * Hank**2 + 9.71 * Hank * Dale + 8.11 * Hank * Jean +
               4.03 * Dale * Jean + 7.08 * Dale * George + 1.55 * George**2 +
               3.05 * Hank + 5.16 * Jean, GRB.MINIMIZE)

# Constraints
m.addConstr(Hank >= 0) # No lower bound explicitly given for hours worked

# Paperwork competence rating constraints
m.addConstr(1 * Hank**2 + 11 * Jean**2 >= 41)
m.addConstr(6 * Dale + 11 * Jean >= 41)
m.addConstr(6 * Dale + 6 * George >= 41)
m.addConstr(Hank**2 + 6 * Dale**2 >= 29)
m.addConstr(Hank**2 + 6 * Dale**2 + 11 * Jean**2 >= 26)
m.addConstr(1 * Hank + 11 * Jean + 6 * George >= 26)
m.addConstr(1 * Hank + 6 * Dale + 11 * Jean >= 43)
m.addConstr(Hank + 11 * Jean + 6 * George >= 43)
m.addConstr(Hank + 6 * Dale + 11 * Jean + 6 * George >= 43)

# Organization score constraints
m.addConstr(23 * Hank + 11 * Dale >= 30)
m.addConstr(5 * Jean + 16 * George >= 29)
m.addConstr((23 * Hank)**2 + (5 * Jean)**2 >= 21)
m.addConstr(23 * Hank + 11 * Dale + 5 * Jean + 16 * George >= 21)

# Work quality rating constraints
m.addConstr(22 * Hank + 2 * George >= 26)
m.addConstr(22 * Hank + 5 * Dale + 14 * Jean + 2 * George >= 26)

# Dollar cost per hour constraints
m.addConstr(13 * Dale + 7 * Jean >= 47)
m.addConstr((19 * Hank)**2 + (13 * Dale)**2 >= 19)
m.addConstr((7 * Jean)**2 + (22 * George)**2 >= 39)
m.addConstr(13 * Dale + 22 * George >= 18)
m.addConstr(19 * Hank + 13 * Dale + 7 * Jean + 22 * George >= 18)

# Upper bound constraints
m.addConstr(Hank**2 + 11 * Jean**2 <= 165)
m.addConstr(6 * Dale**2 + 11 * Jean**2 + 6 * George**2 <= 178)
m.addConstr(23 * Hank + 11 * Dale + 16 * George <= 137)
m.addConstr((23 * Hank)**2 + (5 * Jean)**2 + (16 * George)**2 <= 88)
m.addConstr(22 * Hank + 2 * George <= 121)
m.addConstr(14 * Jean + 2 * George <= 144)
m.addConstr((5 * Dale)**2 + (2 * George)**2 <= 170)
m.addConstr((19 * Hank)**2 + (13 * Dale)**2 + (22 * George)**2 <= 90)
m.addConstr((13 * Dale)**2 + (7 * Jean)**2 + (22 * George)**2 <= 55)
m.addConstr(19 * Hank + 13 * Dale + 7 * Jean <= 146)
m.addConstr(19 * Hank + 7 * Jean + 22 * George <= 129)

# Solve the model
m.optimize()

# Print results if solution was found, otherwise print a message indicating no feasible solution exists
if m.status == GRB.OPTIMAL:
    print("Solution found:")
    for v in m.getVars():
        print(f"{v.varName}: {v.x}")
else:
    print("No feasible solution exists.")
```