## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'chicken drumsticks' and 'chicken thighs', which we can denote as $x_1$ and $x_2$ respectively. The objective function to maximize is $5.19x_1^2 + 1.48x_1x_2 + 2.41x_2^2 + 1.82x_1 + 9.63x_2$.

## Step 2: List the constraints in symbolic notation
The constraints given are:
1. $3x_1 + 8x_2 \geq 53$
2. $11^2x_1^2 + 6^2x_2^2 \geq 24$
3. $2^2x_1^2 + 6^2x_2^2 \geq 35$
4. $9x_1 + 7x_2 \geq 27$
5. $11x_1 + 6x_2 \geq 14$
6. $8x_1^2 - 5x_2^2 \geq 0$
7. $3x_1 + 8x_2 \leq 108$
8. $11x_1 + 6x_2 \leq 73$
9. $2x_1 + 6x_2 \leq 84$
10. $9x_1 + 7x_2 \leq 85$
11. $11^2x_1^2 + 6^2x_2^2 \leq 47^2$ is incorrect based on the problem description, it should be $11x_1 + 6x_2 \leq 47$
12. $x_1$ and $x_2$ are integers.

## 3: Correct and simplify constraints based on the problem description
Correcting and simplifying:
- The actual constraint for iron is $11x_1 + 6x_2 \leq 72$ based on the description provided, not $47^2$.

## 4: Provide the symbolic representation
```json
{
'sym_variables': [('x1', 'chicken drumsticks'), ('x2', 'chicken thighs')],
'objective_function': '5.19*x1^2 + 1.48*x1*x2 + 2.41*x2^2 + 1.82*x1 + 9.63*x2',
'constraints': [
    '3*x1 + 8*x2 >= 53',
    '121*x1^2 + 36*x2^2 >= 24',
    '4*x1^2 + 36*x2^2 >= 35',
    '9*x1 + 7*x2 >= 27',
    '11*x1 + 6*x2 >= 14',
    '8*x1^2 - 5*x2^2 >= 0',
    '3*x1 + 8*x2 <= 108',
    '11*x1 + 6*x2 <= 73',
    '2*x1 + 6*x2 <= 84',
    '9*x1 + 7*x2 <= 85',
    '11*x1 + 6*x2 <= 72',
    'x1 >= 0 and x1 == int(x1)',
    'x2 >= 0 and x2 == int(x2)'
]
}
```

## 5: Write the Gurobi code
```python
import gurobi

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

    # Objective function
    model.setObjective(5.19*x1**2 + 1.48*x1*x2 + 2.41*x2**2 + 1.82*x1 + 9.63*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(3*x1 + 8*x2 >= 53)
    model.addConstr(121*x1**2 + 36*x2**2 >= 24)
    model.addConstr(4*x1**2 + 36*x2**2 >= 35)
    model.addConstr(9*x1 + 7*x2 >= 27)
    model.addConstr(11*x1 + 6*x2 >= 14)
    model.addConstr(8*x1**2 - 5*x2**2 >= 0)
    model.addConstr(3*x1 + 8*x2 <= 108)
    model.addConstr(11*x1 + 6*x2 <= 73)
    model.addConstr(2*x1 + 6*x2 <= 84)
    model.addConstr(9*x1 + 7*x2 <= 85)
    model.addConstr(11*x1 + 6*x2 <= 72)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Chicken drumsticks: ", x1.varValue)
        print("Chicken thighs: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```