## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by George' and 'hours worked by Dale', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $5.47x_1^2 + 1.32x_1x_2 + 4.27x_2^2 + 3.04x_1 + 1.6x_2$.

## 3: List the constraints in symbolic notation
The constraints are:
1. $6x_1 + 15x_2 \geq 37$
2. $-3x_1 + 5x_2 \geq 0$
3. $6^2x_1^2 + 15^2x_2^2 \leq 44$
4. $x_1 + x_2 \leq 44$
5. $x_1$ is a non-integer (continuous) variable
6. $x_2$ is an integer variable

## 4: Correct and simplify constraints based on given resources/attributes
Given {'r0': {'description': 'dollar cost per hour', 'upper_bound': 89, 'x0': 6, 'x1': 15}}, we note:
- The dollar cost per hour for George ($x_0$) is 6.
- The dollar cost per hour for Dale ($x_1$) is 15.
The third constraint should actually reflect the sum of the squares of the costs times the hours worked: $6^2x_1^2 + 15^2x_2^2 \leq 44$.

## 5: Write down the problem in a standard optimization format
Maximize $5.47x_1^2 + 1.32x_1x_2 + 4.27x_2^2 + 3.04x_1 + 1.6x_2$

Subject to:
1. $6x_1 + 15x_2 \geq 37$
2. $-3x_1 + 5x_2 \geq 0$
3. $36x_1^2 + 225x_2^2 \leq 44$
4. $x_1 + x_2 \leq 44$

## 6: Symbolic representation in JSON format
```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': [
'6*x1 + 15*x2 >= 37',
'-3*x1 + 5*x2 >= 0',
'36*x1^2 + 225*x2^2 <= 44',
'x1 + x2 <= 44',
'x2 >= 0', 'x2 <= 44'  # Adding bounds for integer variable x2
]
}
```

## 7: Implement the optimization problem using Gurobi
```python
import gurobi

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

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

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

    # Constraints
    model.addConstr(6*x1 + 15*x2 >= 37, name="cost_constraint")
    model.addConstr(-3*x1 + 5*x2 >= 0, name="work_constraint")
    model.addConstr(36*x1**2 + 225*x2**2 <= 44, name="cost_squared_constraint")
    model.addConstr(x1 + x2 <= 44, name="total_hours_constraint")

    # Optimize the model
    model.optimize()

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

solve_optimization_problem()
```