## Step 1: Define the symbolic representation of the variables
The variables are 'apples', 'eggs', 'hamburgers', and 'rotisserie chickens', which can be represented symbolically as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into symbolic notation
The objective function to minimize is $9x_0^2 + 5x_0x_1 + 3x_0x_3 + 9x_1x_2 + 9x_2^2 + 8x_3^2 + 5x_0 + 6x_1 + 2x_2 + 4x_3$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $10x_2 + 4x_3 \geq 15$
- $20x_1 + 10x_2 \geq 15$
- $20x_1 + 4x_3 \geq 10$
- $x_0 + 4x_3 \geq 26$
- $x_0 + 20x_1 + 10x_2 + 4x_3 \geq 26$
- $3x_0 + 8x_2 \geq 42$
- $6x_1 + x_3 \geq 19$
- $8x_2 + x_3 \geq 19$
- $3x_0 + 6x_1 \geq 43$
- $6x_1^2 + 8x_2^2 + x_3^2 \geq 52$
- $3x_0^2 + 6x_1^2 + 8x_2^2 \geq 52$
- $6x_1^2 + 8x_2^2 + x_3^2 \geq 41$
- $3x_0 + 6x_1 + 8x_2 \geq 41$
- $3x_0 + 6x_1 + 8x_2 + x_3 \geq 41$
- $x_0^2 + x_3^2 \leq 89$
- $x_0^2 + 20^2x_1^2 + 10^2x_2^2 \leq 58$
- $6x_1 + 8x_2 \leq 72$
- $3x_0 + 8x_2 + x_3 \leq 101$
- $3x_0^2 + 6x_1^2 + 8x_2^2 \leq 89$
- $3x_0^2 + 6x_1^2 + x_3^2 \leq 172$

## 4: Create the symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'apples'), ('x1', 'eggs'), ('x2', 'hamburgers'), ('x3', 'rotisserie chickens')],
    'objective_function': '9*x0^2 + 5*x0*x1 + 3*x0*x3 + 9*x1*x2 + 9*x2^2 + 8*x3^2 + 5*x0 + 6*x1 + 2*x2 + 4*x3',
    'constraints': [
        '10*x2 + 4*x3 >= 15',
        '20*x1 + 10*x2 >= 15',
        '20*x1 + 4*x3 >= 10',
        'x0 + 4*x3 >= 26',
        'x0 + 20*x1 + 10*x2 + 4*x3 >= 26',
        '3*x0 + 8*x2 >= 42',
        '6*x1 + x3 >= 19',
        '8*x2 + x3 >= 19',
        '3*x0 + 6*x1 >= 43',
        '6*x1^2 + 8*x2^2 + x3^2 >= 52',
        '3*x0^2 + 6*x1^2 + 8*x2^2 >= 52',
        '6*x1^2 + 8*x2^2 + x3^2 >= 41',
        '3*x0 + 6*x1 + 8*x2 >= 41',
        '3*x0 + 6*x1 + 8*x2 + x3 >= 41',
        'x0^2 + x3^2 <= 89',
        'x0^2 + 400*x1^2 + 100*x2^2 <= 58',
        '6*x1 + 8*x2 <= 72',
        '3*x0 + 8*x2 + x3 <= 101',
        '3*x0^2 + 6*x1^2 + 8*x2^2 <= 89',
        '3*x0^2 + 6*x1^2 + x3^2 <= 172'
    ]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi

# Define the model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="apples", lb=-gurobi.GRB.INFINITY)
x1 = m.addVar(name="eggs", lb=-gurobi.GRB.INFINITY)
x2 = m.addVar(name="hamburgers", lb=-gurobi.GRB.INFINITY)
x3 = m.addVar(name="rotisserie_chickens", lb=-gurobi.GRB.INFINITY)

# Objective function
m.setObjective(9*x0**2 + 5*x0*x1 + 3*x0*x3 + 9*x1*x2 + 9*x2**2 + 8*x3**2 + 5*x0 + 6*x1 + 2*x2 + 4*x3, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(10*x2 + 4*x3 >= 15)
m.addConstr(20*x1 + 10*x2 >= 15)
m.addConstr(20*x1 + 4*x3 >= 10)
m.addConstr(x0 + 4*x3 >= 26)
m.addConstr(x0 + 20*x1 + 10*x2 + 4*x3 >= 26)
m.addConstr(3*x0 + 8*x2 >= 42)
m.addConstr(6*x1 + x3 >= 19)
m.addConstr(8*x2 + x3 >= 19)
m.addConstr(3*x0 + 6*x1 >= 43)
m.addConstr(6*x1**2 + 8*x2**2 + x3**2 >= 52)
m.addConstr(3*x0**2 + 6*x1**2 + 8*x2**2 >= 52)
m.addConstr(6*x1**2 + 8*x2**2 + x3**2 >= 41)
m.addConstr(3*x0 + 6*x1 + 8*x2 >= 41)
m.addConstr(3*x0 + 6*x1 + 8*x2 + x3 >= 41)
m.addConstr(x0**2 + x3**2 <= 89)
m.addConstr(x0**2 + 400*x1**2 + 100*x2**2 <= 58)
m.addConstr(6*x1 + 8*x2 <= 72)
m.addConstr(3*x0 + 8*x2 + x3 <= 101)
m.addConstr(3*x0**2 + 6*x1**2 + 8*x2**2 <= 89)
m.addConstr(3*x0**2 + 6*x1**2 + x3**2 <= 172)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Apples: ", x0.varValue)
    print("Eggs: ", x1.varValue)
    print("Hamburgers: ", x2.varValue)
    print("Rotisserie Chickens: ", x3.varValue)
else:
    print("The model is infeasible")
```