To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify or clarify the objective function and constraints provided. The objective is to maximize:

\[ 5.47 \times (\text{hours worked by George})^2 + 1.32 \times (\text{hours worked by George}) \times (\text{hours worked by Dale}) + 4.27 \times (\text{hours worked by Dale})^2 + 3.04 \times (\text{hours worked by George}) + 1.6 \times (\text{hours worked by Dale}) \]

Subject to the constraints:

1. The dollar cost per hour for George is $6, and for Dale is $15.
2. The total combined dollar cost per hour from hours worked by both must be at least $37.
3. \(-3 \times (\text{hours worked by George}) + 5 \times (\text{hours worked by Dale}) \geq 0\).
4. The sum of the squared dollar costs per hour for both workers should not exceed $44.
5. The total combined dollar cost per hour from hours worked by both must not exceed $44.
6. Hours worked by George can be non-integer, but hours worked by Dale must be an integer.

Let's denote:
- \(G\) as the hours worked by George,
- \(D\) as the hours worked by Dale.

Given this, we can express the objective function and constraints in a more formal mathematical notation:

### Objective Function
Maximize: \(5.47G^2 + 1.32GD + 4.27D^2 + 3.04G + 1.6D\)

### Constraints
1. \(6G + 15D \geq 37\) (Combined dollar cost per hour minimum)
2. \(-3G + 5D \geq 0\) (Linear constraint on hours worked)
3. \(6^2G^2 + 15^2D^2 \leq 44\) (Squared dollar costs sum constraint, misinterpreted initially as it directly translates to a very tight or possibly infeasible condition due to the squaring of already large numbers; however, sticking strictly to the problem statement)
4. \(6G + 15D \leq 44\) (Combined dollar cost per hour maximum)
5. \(G \in \mathbb{R}\), \(D \in \mathbb{Z}\) (Nature of variables)

Given these constraints and objective function, here's how we can model this problem in Gurobi using Python:

```python
from gurobipy import *

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

# Define the variables
G = m.addVar(vtype=GRB.CONTINUOUS, name="Hours_Worked_by_George")
D = m.addVar(vtype=GRB.INTEGER, name="Hours_Worked_by_Dale")

# Objective function
m.setObjective(5.47*G**2 + 1.32*G*D + 4.27*D**2 + 3.04*G + 1.6*D, GRB.MAXIMIZE)

# Constraints
m.addConstr(6*G + 15*D >= 37, name="Minimum_Combined_Dollar_Cost")
m.addConstr(-3*G + 5*D >= 0, name="Linear_Constraint_on_Hours")
m.addConstr(36*G**2 + 225*D**2 <= 44, name="Squared_Dollar_Costs_Sum") # Corrected constraint based on initial misinterpretation
m.addConstr(6*G + 15*D <= 44, name="Maximum_Combined_Dollar_Cost")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by George: {G.x}")
    print(f"Hours worked by Dale: {D.x}")
    print(f"Objective value: {m.objVal}")
else:
    print("No optimal solution found")
```