To convert the given problem into a Gurobi optimization model, we first need to understand and possibly reformulate the objective function and constraints provided in a mathematical form that can be directly translated into Gurobi's Python API.

The objective function to minimize 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. Umami index of green beans is 3.
2. Umami index of cheeseburgers is 4.
3. Total combined umami index from green beans squared and cheeseburgers squared must be at least 13: \(3^2 \times (\text{green beans}) + 4^2 \times (\text{cheeseburgers}) \geq 13\).
4. Total combined umami index from green beans plus cheeseburgers should be at minimum 13: \(3 \times (\text{green beans}) + 4 \times (\text{cheeseburgers}) \geq 13\).
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 44 at maximum: \(3 \times (\text{green beans}) + 4 \times (\text{cheeseburgers}) \leq 44\).
7. There must be a whole number amount of green beans.
8. The amount of cheeseburgers can be non-integer.

Given these conditions, we'll create the model using Gurobi's Python interface. Note that for constraints involving squared terms directly in inequalities (like constraint 3), we need to ensure these are correctly represented as they might not directly translate into linear constraints which Gurobi handles natively. However, given the nature of this problem, a direct formulation considering all aspects seems feasible.

```python
from gurobipy import *

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

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

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

# Constraints
m.addConstr(9 * green_beans + 16 * cheeseburgers >= 13, name="umami_index_squared_min")
m.addConstr(3 * green_beans + 4 * cheeseburgers >= 13, name="umami_index_min")
m.addConstr(-5 * green_beans**2 + 2 * cheeseburgers**2 >= 0, name="green_beans_cheeseburgers_relationship")
m.addConstr(3 * green_beans + 4 * cheeseburgers <= 44, name="umami_index_max")

# Optimizer parameters
m.Params.OutputFlag = 1

# Solve the model
m.optimize()

```