To solve this optimization problem, we need to translate the given natural language description into a mathematical model and then implement it using Gurobi. The objective function to maximize is:

\[8.83 \times (\text{fruit salads})^2 + 6.48 \times (\text{fruit salads}) \times (\text{potatoes}) + 3.67 \times (\text{fruit salads}) + 6.4 \times (\text{potatoes})\]

Subject to the constraints:

1. Fat from fruit salads: \(12 \times (\text{fruit salads})\)
2. Iron from fruit salads: \(8 \times (\text{fruit salads})\)
3. Fat from potatoes: \(8 \times (\text{potatoes})\)
4. Iron from potatoes: \(15 \times (\text{potatoes})\)
5. Minimum fat requirement: \(12 \times (\text{fruit salads}) + 8 \times (\text{potatoes}) \geq 25\)
6. Minimum iron requirement: \((\text{fruit salads})^2 \times 8 + (\text{potatoes})^2 \times 15 \geq 52\)
7. Non-negativity constraint for a quadratic expression: \(1 \times (\text{fruit salads})^2 - 10 \times (\text{potatoes})^2 \geq 0\)
8. Maximum fat from squared quantities: \((\text{fruit salads})^2 \times 12 + (\text{potatoes})^2 \times 8 \leq 63\)
9. Maximum fat: \(12 \times (\text{fruit salads}) + 8 \times (\text{potatoes}) \leq 63\)
10. Maximum iron: \(8 \times (\text{fruit salads}) + 15 \times (\text{potatoes}) \leq 141\)

Given that the quantity of fruit salads does not have to be whole but the number of potatoes must be a whole number, we'll define "fruit salads" as a continuous variable and "potatoes" as an integer variable.

```python
from gurobipy import *

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

# Define variables
fruit_salads = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="fruit_salads")
potatoes = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.INTEGER, name="potatoes")

# Objective function
m.setObjective(8.83 * fruit_salads**2 + 6.48 * fruit_salads * potatoes + 3.67 * fruit_salads + 6.4 * potatoes, GRB.MAXIMIZE)

# Constraints
m.addConstr(12*fruit_salads + 8*potatoes >= 25, name="min_fat")
m.addConstr((fruit_salads**2)*8 + (potatoes**2)*15 >= 52, name="min_iron_squared")
m.addConstr(fruit_salads**2 - 10*potatoes**2 >= 0, name="quadratic_non_negativity")
m.addConstr((fruit_salads**2)*12 + (potatoes**2)*8 <= 63, name="max_fat_squared")
m.addConstr(12*fruit_salads + 8*potatoes <= 63, name="max_fat")
m.addConstr(8*fruit_salads + 15*potatoes <= 141, name="max_iron")

# Solve the model
m.optimize()

print("Objective Function Value:", m.objVal)
for v in m.getVars():
    print(f"{v.varName}: {v.x}")
```