## Problem Description and Formulation

The problem requires minimizing an objective function subject to several constraints. The objective function to be minimized is:

\[ 8.05 \times (\text{green beans})^2 + 7.14 \times \text{green beans} \times \text{cheeseburgers} + 7.69 \times (\text{cheeseburgers})^2 + 2.38 \times \text{green beans} \]

The constraints are:

1. The umami index of green beans is 3.
2. The umami index of cheeseburgers is 4.
3. The total combined umami index from green beans and cheeseburgers squared must be at least 13.
4. The total combined umami index from green beans and cheeseburgers should be at minimum 13.
5. \(-5 \times (\text{green beans})^2 + 2 \times (\text{cheeseburgers})^2 \geq 0\).
6. The total combined umami index from green beans and cheeseburgers has to be at maximum 44.
7. There must be a whole number amount of green beans.
8. You can have a non-integer amount of cheeseburgers.

## Gurobi Code Formulation

```python
import gurobi

# Create a new Gurobi model
m = gurobi.Model()

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

# Objective function
m.setObjective(8.05 * green_beans ** 2 + 7.14 * green_beans * cheeseburgers + 7.69 * cheeseburgers ** 2 + 2.38 * green_beans, gurobi.GRB.MINIMIZE)

# Constraints
# 1. Umami index of green beans is 3 (not needed as it's a variable property)
# 2. Umami index of cheeseburgers is 4 (not needed as it's a variable property)
# 3. Total combined umami index from green beans and cheeseburgers squared must be at least 13
m.addConstraint(3 ** 2 * green_beans ** 2 + 4 ** 2 * cheeseburgers ** 2 >= 13, name="umami_squared_min")
# 4. Total combined umami index from green beans and cheeseburgers should be at minimum 13
m.addConstraint(3 * green_beans + 4 * cheeseburgers >= 13, name="umami_min")
# 5. -5 * (green beans)^2 + 2 * (cheeseburgers)^2 >= 0
m.addConstraint(-5 * green_beans ** 2 + 2 * cheeseburgers ** 2 >= 0, name="cheeseburgers_vs_green_beans")
# 6. Total combined umami index from green beans and cheeseburgers has to be at maximum 44
m.addConstraint(3 * green_beans + 4 * cheeseburgers <= 44, name="umami_max")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Green beans: {green_beans.varValue}")
    print(f"Cheeseburgers: {cheeseburgers.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```