## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and the objective function based on the given problem description.

- Let $x_1$ represent 'green beans' and $x_2$ represent 'cheeseburgers'.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $8.05x_1^2 + 7.14x_1x_2 + 7.69x_2^2 + 2.38x_1$.

## 3: Define the constraints in symbolic notation
The constraints are:
1. $3x_1^2 + 4x_2^2 \geq 13$
2. $3x_1 + 4x_2 \geq 13$
3. $-5x_1^2 + 2x_2^2 \geq 0$
4. $3x_1 + 4x_2 \leq 44$
5. $x_1$ is an integer
6. $x_2$ is a real number

## 4: Create a symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'green beans'), ('x2', 'cheeseburgers')],
    'objective_function': '8.05*x1^2 + 7.14*x1*x2 + 7.69*x2^2 + 2.38*x1',
    'constraints': [
        '3*x1^2 + 4*x2^2 >= 13',
        '3*x1 + 4*x2 >= 13',
        '-5*x1^2 + 2*x2^2 >= 0',
        '3*x1 + 4*x2 <= 44',
        'x1 is an integer'
    ]
}
```

## Step 5: Implement the problem using Gurobi
Now, let's implement this problem using Gurobi in Python.

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name='green_beans', vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name='cheeseburgers')

    # Set the objective function
    model.setObjective(8.05 * x1**2 + 7.14 * x1 * x2 + 7.69 * x2**2 + 2.38 * x1, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(3 * x1**2 + 4 * x2**2 >= 13)
    model.addConstr(3 * x1 + 4 * x2 >= 13)
    model.addConstr(-5 * x1**2 + 2 * x2**2 >= 0)
    model.addConstr(3 * x1 + 4 * x2 <= 44)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('Green beans: ', x1.varValue)
        print('Cheeseburgers: ', x2.varValue)
    else:
        print('The model is infeasible')

solve_optimization_problem()
```