## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- $x_0$ represents 'packs of napkins'
- $x_1$ represents 'lightbulbs'
- $x_2$ represents 'cookies'
- $x_3$ represents 'paper towel rolls'
- $x_4$ represents 'cartons of milk'

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is: 
$4x_0^2 + 7x_0x_1 + 2x_0x_2 + 4x_0x_3 + 6x_0x_4 + 3x_1^2 + 7x_1x_2 + 9x_1x_3 + 5x_1x_4 + x_2^2 + 7x_2x_3 + 5x_3^2 + 9x_3x_4 + 6x_4^2 + 2x_0 + 2x_1 + 7x_2 + 9x_3 + 2x_4$

## Step 3: Define the constraints in symbolic notation
The constraints are as follows:
- $9x_0 \leq 121$ (storage space constraint for packs of napkins)
- $11x_0 \leq 154$ (weight constraint for packs of napkins)
- $3x_1 \leq 121$ (storage space constraint for lightbulbs)
- $11x_1 \leq 154$ (weight constraint for lightbulbs)
- $11x_2 \leq 121$ (storage space constraint for cookies)
- $2x_2 \leq 154$ (weight constraint for cookies)
- $4x_3 \leq 121$ (storage space constraint for paper towel rolls)
- $14x_3 \leq 154$ (weight constraint for paper towel rolls)
- $8x_4 \leq 121$ (storage space constraint for cartons of milk)
- $2x_4 \leq 154$ (weight constraint for cartons of milk)
- $4x_3 + 8x_4 \geq 13$ (storage space constraint for paper towel rolls and cartons of milk)
- $9x_0 + 11x_2 \geq 21$ (storage space constraint for packs of napkins and cookies)
- $9x_0 + 3x_1 \geq 23$ (storage space constraint for packs of napkins and lightbulbs)
- $3x_1 + 4x_3 \geq 11$ (storage space constraint for lightbulbs and paper towel rolls)
- $3x_1^2 + 2x_4^2 \geq 23$ (storage space constraint for lightbulbs squared and cartons of milk squared)
- $9x_0 + 4x_3 \geq 15$ (storage space constraint for packs of napkins and paper towel rolls)
- $9x_0 + 3x_1 + 4x_3 \geq 12$ (storage space constraint for packs of napkins, lightbulbs, and paper towel rolls)
- $9x_0 + 3x_1 + 11x_2 + 4x_3 + 8x_4 \geq 12$ (storage space constraint for all variables)
- $11x_0^2 + 11x_1^2 \geq 30$ (weight constraint for packs of napkins squared and lightbulbs squared)
- $11x_0^2 + 2x_4^2 \geq 14$ (weight constraint for packs of napkins squared and cartons of milk squared)
- $11x_0^2 + 2x_2^2 \geq 16$ (weight constraint for packs of napkins squared and cookies squared)
- $11x_1 + 2x_4 \geq 16$ (weight constraint for lightbulbs and cartons of milk)
- $11x_1 + 14x_3 \geq 23$ (weight constraint for lightbulbs and paper towel rolls)
- $2x_2 + 2x_4 \geq 27$ (weight constraint for cookies and cartons of milk)
- $2x_2^2 + 5x_3^2 \geq 11$ (weight constraint for cookies squared and paper towel rolls squared)
- $11x_0 + 14x_3 \geq 23$ (weight constraint for packs of napkins and paper towel rolls)
- $11x_0^2 + 11x_1^2 + 2x_4^2 \geq 27$ (weight constraint for packs of napkins squared, lightbulbs squared, and cartons of milk squared)
- $2x_2 + 14x_3 + 2x_4 \geq 27$ (weight constraint for cookies, paper towel rolls, and cartons of milk)
- $11x_0 + 11x_1 + 2x_4 \geq 22$ (weight constraint for packs of napkins, lightbulbs, and cartons of milk)
- $2x_2 + 14x_3 + 2x_4 \geq 22$ (weight constraint for cookies, paper towel rolls, and cartons of milk)
- $11x_0 + 11x_1 + 2x_2 + 14x_3 + 2x_4 \geq 22$ (weight constraint for all variables)
- $-x_1 + 4x_4 \geq 0$ (linear constraint)
- $-8x_2 + 8x_4 \geq 0$ (linear constraint)
- $3x_1 + 11x_2 \leq 121$ (storage space constraint for lightbulbs and cookies)
- $9x_0^2 + 11x_2^2 \leq 37$ (storage space constraint for packs of napkins squared and cookies squared)

## 4: Create the Gurobi model and variables
We will now create the Gurobi model and variables.

```python
import gurobi

# Create a new Gurobi model
model = gurobi.Model()

# Define the variables
x0 = model.addVar(name='x0', vtype=gurobi.GRB.INTEGER) # packs of napkins
x1 = model.addVar(name='x1', vtype=gurobi.GRB.INTEGER) # lightbulbs
x2 = model.addVar(name='x2', vtype=gurobi.GRB.INTEGER) # cookies
x3 = model.addVar(name='x3', vtype=gurobi.GRB.INTEGER) # paper towel rolls
x4 = model.addVar(name='x4', vtype=gurobi.GRB.INTEGER) # cartons of milk

# Objective function
model.setObjective(4*x0**2 + 7*x0*x1 + 2*x0*x2 + 4*x0*x3 + 6*x0*x4 + 
                   3*x1**2 + 7*x1*x2 + 9*x1*x3 + 5*x1*x4 + x2**2 + 
                   7*x2*x3 + 5*x3**2 + 9*x3*x4 + 6*x4**2 + 2*x0 + 
                   2*x1 + 7*x2 + 9*x3 + 2*x4, gurobi.GRB.MINIMIZE)

# Constraints
model.addConstr(9*x0 <= 121)
model.addConstr(11*x0 <= 154)
model.addConstr(3*x1 <= 121)
model.addConstr(11*x1 <= 154)
model.addConstr(11*x2 <= 121)
model.addConstr(2*x2 <= 154)
model.addConstr(4*x3 <= 121)
model.addConstr(14*x3 <= 154)
model.addConstr(8*x4 <= 121)
model.addConstr(2*x4 <= 154)

model.addConstr(4*x3 + 8*x4 >= 13)
model.addConstr(9*x0 + 11*x2 >= 21)
model.addConstr(9*x0 + 3*x1 >= 23)
model.addConstr(3*x1 + 4*x3 >= 11)
model.addConstr(3*x1**2 + 2*x4**2 >= 23)
model.addConstr(9*x0 + 4*x3 >= 15)
model.addConstr(9*x0 + 3*x1 + 4*x3 >= 12)
model.addConstr(9*x0 + 3*x1 + 11*x2 + 4*x3 + 8*x4 >= 12)

model.addConstr(11*x0**2 + 11*x1**2 >= 30)
model.addConstr(11*x0**2 + 2*x4**2 >= 14)
model.addConstr(11*x0**2 + 2*x2**2 >= 16)
model.addConstr(11*x1 + 2*x4 >= 16)
model.addConstr(11*x1 + 14*x3 >= 23)
model.addConstr(2*x2 + 2*x4 >= 27)
model.addConstr(2*x2**2 + 5*x3**2 >= 11)
model.addConstr(11*x0 + 14*x3 >= 23)
model.addConstr(11*x0**2 + 11*x1**2 + 2*x4**2 >= 27)
model.addConstr(2*x2 + 14*x3 + 2*x4 >= 27)
model.addConstr(11*x0 + 11*x1 + 2*x4 >= 22)
model.addConstr(2*x2 + 14*x3 + 2*x4 >= 22)
model.addConstr(11*x0 + 11*x1 + 2*x2 + 14*x3 + 2*x4 >= 22)

model.addConstr(-x1 + 4*x4 >= 0)
model.addConstr(-8*x2 + 8*x4 >= 0)
model.addConstr(3*x1 + 11*x2 <= 121)
model.addConstr(9*x0**2 + 11*x2**2 <= 37)

# Solve the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', model.objval)
    print('x0: ', x0.varValue)
    print('x1: ', x1.varValue)
    print('x2: ', x2.varValue)
    print('x3: ', x3.varValue)
    print('x4: ', x4.varValue)
else:
    print('No solution found')
```

### Symbolic Representation

```json
{
    'sym_variables': [
        ['x0', 'packs of napkins'],
        ['x1', 'lightbulbs'],
        ['x2', 'cookies'],
        ['x3', 'paper towel rolls'],
        ['x4', 'cartons of milk']
    ],
    'objective_function': '4*x0^2 + 7*x0*x1 + 2*x0*x2 + 4*x0*x3 + 6*x0*x4 + 3*x1^2 + 7*x1*x2 + 9*x1*x3 + 5*x1*x4 + x2^2 + 7*x2*x3 + 5*x3^2 + 9*x3*x4 + 6*x4^2 + 2*x0 + 2*x1 + 7*x2 + 9*x3 + 2*x4',
    'constraints': [
        '9*x0 <= 121',
        '11*x0 <= 154',
        '3*x1 <= 121',
        '11*x1 <= 154',
        '11*x2 <= 121',
        '2*x2 <= 154',
        '4*x3 <= 121',
        '14*x3 <= 154',
        '8*x4 <= 121',
        '2*x4 <= 154',
        '4*x3 + 8*x4 >= 13',
        '9*x0 + 11*x2 >= 21',
        '9*x0 + 3*x1 >= 23',
        '3*x1 + 4*x3 >= 11',
        '3*x1^2 + 2*x4^2 >= 23',
        '9*x0 + 4*x3 >= 15',
        '9*x0 + 3*x1 + 4*x3 >= 12',
        '9*x0 + 3*x1 + 11*x2 + 4*x3 + 8*x4 >= 12',
        '11*x0^2 + 11*x1^2 >= 30',
        '11*x0^2 + 2*x4^2 >= 14',
        '11*x0^2 + 2*x2^2 >= 16',
        '11*x1 + 2*x4 >= 16',
        '11*x1 + 14*x3 >= 23',
        '2*x2 + 2*x4 >= 27',
        '2*x2^2 + 5*x3^2 >= 11',
        '11*x0 + 14*x3 >= 23',
        '11*x0^2 + 11*x1^2 + 2*x4^2 >= 27',
        '2*x2 + 14*x3 + 2*x4 >= 27',
        '11*x0 + 11*x1 + 2*x4 >= 22',
        '2*x2 + 14*x3 + 2*x4 >= 22',
        '11*x0 + 11*x1 + 2*x2 + 14*x3 + 2*x4 >= 22',
        '-x1 + 4*x4 >= 0',
        '-8*x2 + 8*x4 >= 0',
        '3*x1 + 11*x2 <= 121',
        '9*x0^2 + 11*x2^2 <= 37'
    ]
}
```