To tackle this problem, we first need to break down the given natural language description into a symbolic representation of the optimization problem. This involves defining variables, an objective function, and constraints based on the provided information.

### Symbolic Representation

Let's define:
- $x_1$ as 'hours worked by George'
- $x_2$ as 'hours worked by Dale'

The objective function is to maximize: 
\[5.47x_1^2 + 1.32x_1x_2 + 4.27x_2^2 + 3.04x_1 + 1.6x_2\]

Given constraints:
1. George's dollar cost per hour is $6, and Dale's is $15.
2. The total combined dollar cost per hour from hours worked by George and Dale should be at least $37.
3. \(-3x_1 + 5x_2 \geq 0\)
4. The total combined dollar cost per hour from hours worked by George squared plus hours worked by Dale squared must be 44 or less, but this seems to actually refer to the costs themselves, not their squares. Thus, it's interpreted as \(6x_1 + 15x_2 \leq 44\).
5. The constraint about the total combined dollar cost per hour from hours worked by George and Dale being at most 44 is already covered in point 4.

The symbolic representation of the problem can be summarized as:
```json
{
  'sym_variables': [('x1', 'hours worked by George'), ('x2', 'hours worked by Dale')],
  'objective_function': '5.47*x1**2 + 1.32*x1*x2 + 4.27*x2**2 + 3.04*x1 + 1.6*x2',
  'constraints': [
    '-3*x1 + 5*x2 >= 0', 
    '6*x1 + 15*x2 >= 37', 
    '6*x1 + 15*x2 <= 44'
  ]
}
```

### Gurobi Code

To solve this problem using Gurobi, we'll write the Python code to implement the model. Note that since $x_2$ (hours worked by Dale) must be an integer, we'll use a mixed-integer quadratic program (MIQP).

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="hours_worked_by_George")
x2 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.INTEGER, name="hours_worked_by_Dale")

# Objective function
m.setObjective(5.47*x1**2 + 1.32*x1*x2 + 4.27*x2**2 + 3.04*x1 + 1.6*x2, GRB.MAXIMIZE)

# Constraints
m.addConstr(-3*x1 + 5*x2 >= 0, name="constraint_1")
m.addConstr(6*x1 + 15*x2 >= 37, name="constraint_2")
m.addConstr(6*x1 + 15*x2 <= 44, name="constraint_3")

# Optimize model
m.optimize()

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