## Step 1: Define the symbolic representation of the variables
The variables are 'cornichons' and 'peanutbutter sandwiches', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $4x_1^2 + 6x_1x_2 + 3x_2^2 + 4x_1 + 2x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $11x_1 + 10x_2 \geq 22$ (at least 22 grams of fat)
- $5x_1 + 8x_2 \geq 21$ (spend at least $21)
- $6x_1^2 - 3x_2^2 \geq 0$
- $(11x_1 + 10x_2)^2 \leq 62^2$ is not linear and seems to be misinterpreted. The correct interpretation seems to be $11x_1 + 10x_2 \leq 62$ (at most 62 grams of fat)
- $5x_1 + 8x_2 \leq 59$ (spend at most $59)
- $x_1, x_2$ are integers (whole number of cornichons and nonfractional number of peanutbutter sandwiches)

## 4: Correct and Linearize Constraints for Clarity
Correcting and clarifying the constraints:
- Fat constraint: $11x_1 + 10x_2 \geq 22$
- Cost constraint: $5x_1 + 8x_2 \geq 21$
- Non-negativity and bounds are implicitly handled by constraints.

## 5: Symbolic Representation
```json
{
'sym_variables': [('x1', 'cornichons'), ('x2', 'peanutbutter sandwiches')],
'objective_function': '4*x1^2 + 6*x1*x2 + 3*x2^2 + 4*x1 + 2*x2',
'constraints': [
    '11*x1 + 10*x2 >= 22',
    '5*x1 + 8*x2 >= 21',
    '6*x1^2 - 3*x2^2 >= 0',
    '11*x1 + 10*x2 <= 62',
    '5*x1 + 8*x2 <= 59',
    'x1, x2 are integers'
]
}
```

## 6: Gurobi Code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x1 = model.addVar(name="cornichons", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="peanutbutter_sandwiches", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(4*x1**2 + 6*x1*x2 + 3*x2**2 + 4*x1 + 2*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(11*x1 + 10*x2 >= 22, name="fat_constraint")
    model.addConstr(5*x1 + 8*x2 >= 21, name="cost_constraint")
    model.addConstr(6*x1**2 - 3*x2**2 >= 0, name="nonlinear_constraint")
    model.addConstr(11*x1 + 10*x2 <= 62, name="fat_bound")
    model.addConstr(5*x1 + 8*x2 <= 59, name="cost_bound")

    # Solve the model
    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Cornichons: {x1.varValue}, Peanutbutter Sandwiches: {x2.varValue}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has a non-optimal status.")

solve_optimization_problem()
```