To tackle this problem, let's first break down the components and translate them into a symbolic representation as requested.

### Symbolic Representation

Given:
- Variables: `x1` for 'green beans', `x2` for 'cheeseburgers'.
- Objective Function: Minimize \(8.05x_1^2 + 7.14x_1x_2 + 7.69x_2^2 + 2.38x_1\).
- Constraints:
  1. Umami index of green beans is 3, and cheeseburgers is 4: Not directly a constraint but part of the umami index calculation.
  2. Total combined umami index from green beans squared and cheeseburgers squared must be at least 13: \(3^2x_1^2 + 4^2x_2^2 \geq 13\).
  3. Total combined umami index from green beans plus cheeseburgers should be at minimum 13: \(3x_1 + 4x_2 \geq 13\).
  4. \(-5x_1^2 + 2x_2^2 \geq 0\).
  5. Total combined umami index from green beans and cheeseburgers has to be at maximum 44: \(3x_1 + 4x_2 \leq 44\).
  6. There must be a whole number amount of green beans: \(x_1 \in \mathbb{Z}\).
  7. You can have a non-integer amount of cheeseburgers.

### Symbolic Problem Description

```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': [
    '9*x1**2 + 16*x2**2 >= 13',
    '3*x1 + 4*x2 >= 13',
    '-5*x1**2 + 2*x2**2 >= 0',
    '3*x1 + 4*x2 <= 44'
  ]
}
```

### Gurobi Code

To solve this problem using Gurobi, we need to formulate it as a Mixed-Integer Quadratic Program (MIQP) because of the integer constraint on `x1` and the quadratic terms in both the objective function and constraints.

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="green_beans")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="cheeseburgers")

# Objective function
m.setObjective(8.05*x1**2 + 7.14*x1*x2 + 7.69*x2**2 + 2.38*x1, GRB.MINIMIZE)

# Constraints
m.addConstr(9*x1**2 + 16*x2**2 >= 13, name="umami_index_squared")
m.addConstr(3*x1 + 4*x2 >= 13, name="total_umami_index_min")
m.addConstr(-5*x1**2 + 2*x2**2 >= 0, name="quadratic_constraint")
m.addConstr(3*x1 + 4*x2 <= 44, name="total_umami_index_max")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Green Beans: {x1.x}")
    print(f"Cheeseburgers: {x2.x}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found")
```