To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. The variables are 'fruit salads' and 'potatoes', which we will represent as $x_1$ and $x_2$, respectively.

The objective function is given by: 
\[8.83(x_1)^2 + 6.48(x_1)(x_2) + 3.67(x_1) + 6.4(x_2)\]

The constraints are:
1. Fat from fruit salads and potatoes should be at least 25 grams: \[12x_1 + 8x_2 \geq 25\]
2. Iron from the squared quantities of fruit salads and potatoes should be at least 52 milligrams: \[(8x_1)^2 + (15x_2)^2 \geq 52\]
3. A constraint involving squares of $x_1$ and $x_2$: \[x_1^2 - 10x_2^2 \geq 0\]
4. The total fat from the squared quantities should not exceed 63 grams: \[(12x_1)^2 + (8x_2)^2 \leq 63\]
5. Total fat from fruit salads and potatoes should not exceed 63 grams: \[12x_1 + 8x_2 \leq 63\]
6. Total iron from fruit salads and potatoes should not exceed 141 milligrams: \[8x_1 + 15x_2 \leq 141\]

Given the problem's requirements, here is the symbolic representation:
```json
{
    'sym_variables': [('x1', 'fruit salads'), ('x2', 'potatoes')],
    'objective_function': '8.83*(x1)**2 + 6.48*x1*x2 + 3.67*x1 + 6.4*x2',
    'constraints': [
        '12*x1 + 8*x2 >= 25',
        '(8*x1)**2 + (15*x2)**2 >= 52',
        'x1**2 - 10*x2**2 >= 0',
        '(12*x1)**2 + (8*x2)**2 <= 63',
        '12*x1 + 8*x2 <= 63',
        '8*x1 + 15*x2 <= 141'
    ]
}
```

Now, let's write the Gurobi code to solve this optimization problem:
```python
from gurobipy import *

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

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

# Objective function
m.setObjective(8.83*x1**2 + 6.48*x1*x2 + 3.67*x1 + 6.4*x2, GRB.MAXIMIZE)

# Constraints
m.addConstr(12*x1 + 8*x2 >= 25)
m.addConstr((8*x1)**2 + (15*x2)**2 >= 52)
m.addConstr(x1**2 - 10*x2**2 >= 0)
m.addConstr((12*x1)**2 + (8*x2)**2 <= 63)
m.addConstr(12*x1 + 8*x2 <= 63)
m.addConstr(8*x1 + 15*x2 <= 141)

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"fruit_salads: {x1.x}, potatoes: {x2.x}")
else:
    print("No optimal solution found.")
```