To address the problem described, we first need to translate the natural language description into a symbolic representation that can be used in an optimization model. The variables and their corresponding objects are as follows:

- `x1`: potatoes
- `x2`: cherry pies
- `x3`: rotisserie chickens
- `x4`: cantaloupes

The objective function to maximize is given by:
6*x1^2 + 7*x1*x3 + 5*x2^2 + 4*x2*x3 + 2*x3^2 + 6*x3*x4 + 5*x4^2 + x1

The constraints based on the carbohydrates content are:
- 13x1 (grams of carbohydrates in potatoes)
- 10x2 (grams of carbohydrates in cherry pies)
- 11x3 (grams of carbohydrates in rotisserie chickens)
- 2x4 (grams of carbohydrates in cantaloupes)

Specific constraints mentioned include:
- 13x1 + 2x4 ≤ 21 (total grams of carbohydrates from potatoes and cantaloupes)
- 13x1 + 10x2 ≤ 46 (total grams of carbohydrates from potatoes and cherry pies)
- 13x1 + 10x2 + 11x3 + 2x4 ≤ 46 (total grams of carbohydrates from all items)

Additionally, x1, x2, x3, and x4 must be integers.

The symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'potatoes'), ('x2', 'cherry pies'), ('x3', 'rotisserie chickens'), ('x4', 'cantaloupes')],
    'objective_function': '6*x1^2 + 7*x1*x3 + 5*x2^2 + 4*x2*x3 + 2*x3^2 + 6*x3*x4 + 5*x4^2 + x1',
    'constraints': [
        '13*x1 + 2*x4 <= 21',
        '13*x1 + 10*x2 <= 46',
        '13*x1 + 10*x2 + 11*x3 + 2*x4 <= 46'
    ]
}
```

Now, let's implement this problem using Gurobi in Python:

```python
from gurobipy import *

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

# Define the variables
x1 = m.addVar(vtype=GRB.INTEGER, name="potatoes")
x2 = m.addVar(vtype=GRB.INTEGER, name="cherry_pies")
x3 = m.addVar(vtype=GRB.INTEGER, name="rotisserie_chickens")
x4 = m.addVar(vtype=GRB.INTEGER, name="cantaloupes")

# Set the objective function
m.setObjective(6*x1**2 + 7*x1*x3 + 5*x2**2 + 4*x2*x3 + 2*x3**2 + 6*x3*x4 + 5*x4**2 + x1, GRB.MAXIMIZE)

# Add constraints
m.addConstr(13*x1 + 2*x4 <= 21, name="Carbs_from_Potatoes_and_Cantaloupes")
m.addConstr(13*x1 + 10*x2 <= 46, name="Carbs_from_Potatoes_and_Cherry_Pies")
m.addConstr(13*x1 + 10*x2 + 11*x3 + 2*x4 <= 46, name="Total_Carbs")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    for v in m.getVars():
        print(f"{v.varName}: {v.x}")
else:
    print("No optimal solution found")
```