## Step 1: Define the symbolic representation of the problem
The problem involves the following variables: 
- packs of paper plates: $x_0$
- bottles of ibuprofen: $x_1$
- toilet paper rolls: $x_2$
- diapers: $x_3$
- cartons of milk: $x_4$

## Step 2: Translate the objective function into algebraic terms
The objective function to minimize is:
\[3.08x_0^2 + 5.45x_0x_1 + 3.4x_0x_2 + 4.05x_0x_3 + 5.91x_0x_4 + 2.46x_1^2 + 9.62x_1x_2 + 8.76x_1x_3 + 2.2x_2^2 + 3.71x_3^2 + 5.58x_4^2 + 5.18x_0 + 7.76x_1 + 3.37x_2 + 5.99x_3 + 8.27x_4\]

## 3: List the constraints in algebraic terms
Constraints:
1. $2x_0 \leq 245$
2. $10x_0 \leq 221$
3. $3x_1 \leq 245$
4. $6x_1 \leq 221$
5. $11x_2 \leq 245$
6. $8x_2 \leq 221$
7. $8x_3 \leq 245$
8. $9x_3 \leq 221$
9. $6x_4 \leq 245$
10. $5x_4 \leq 221$
11. $11x_2^2 + 8x_3^2 \geq 26$
12. $2x_0 + 8x_3 \geq 16$
13. $2x_0 + 6x_4 \geq 18$
14. $2x_0 + 11x_2 \geq 46$
15. $2x_0 + 3x_1 + 11x_2 + 8x_3 + 6x_4 \geq 46$
16. $10x_0^2 + 5x_4^2 \geq 27$
17. $10x_0 + 8x_2 \geq 40$
18. $6x_1^2 + 5x_4^2 \geq 19$
19. $8x_2 + 5x_4 \geq 40$
20. $6x_1 + 9x_3 \geq 42$
21. $10x_0^2 + 8x_2^2 + 9x_3^2 \geq 37$
22. $10x_0 + 6x_1 + 8x_2 + 9x_3 + 5x_4 \geq 37$
23. $-3x_1^2 + 10x_3^2 \geq 0$
24. $3x_1^2 - 4x_2^2 \geq 0$
25. $2x_0^2 + 11x_2^2 + 6x_4^2 \leq 118$
26. $11x_2^2 + 8x_3^2 + 6x_4^2 \leq 111$
27. $2x_0^2 + 3x_1^2 + 6x_4^2 \leq 194$
28. $3x_1^2 + 11x_2^2 + 8x_3^2 \leq 202$
29. $2x_0^2 + 8x_3^2 + 6x_4^2 \leq 203$
30. $2x_0 + 3x_1 + 11x_2 \leq 232$
31. $6x_1 + 9x_3 \leq 109$
32. $6x_1 + 8x_2 + 5x_4 \leq 119$
33. $10x_0 + 6x_1 + 8x_2 \leq 209$
34. $10x_0^2 + 8x_2^2 + 9x_3^2 \leq 153$
35. $8x_2 + 9x_3 + 5x_4 \leq 156$
36. $10x_0 + 6x_1 + 5x_4 \leq 160$
37. $10x_0 + 8x_2 + 5x_4 \leq 182$
38. $6x_1 + 9x_3 + 5x_4 \leq 204$
39. $x_0 \in \mathbb{Z}$
40. $x_1 \in \mathbb{Z}$
41. $x_2 \in \mathbb{Z}$
42. $x_3 \in \mathbb{Z}$
43. $x_4 \in \mathbb{Z}$

## 4: Write the Gurobi code
```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="packs_of_paper_plates", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="bottles_of_ibuprofen", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="toilet_paper_rolls", vtype=gp.GRB.INTEGER)
x3 = m.addVar(name="diapers", vtype=gp.GRB.INTEGER)
x4 = m.addVar(name="cartons_of_milk", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(3.08*x0**2 + 5.45*x0*x1 + 3.4*x0*x2 + 4.05*x0*x3 + 5.91*x0*x4 + 
               2.46*x1**2 + 9.62*x1*x2 + 8.76*x1*x3 + 2.2*x2**2 + 3.71*x3**2 + 
               5.58*x4**2 + 5.18*x0 + 7.76*x1 + 3.37*x2 + 5.99*x3 + 8.27*x4, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(2*x0 <= 245)
m.addConstr(10*x0 <= 221)
m.addConstr(3*x1 <= 245)
m.addConstr(6*x1 <= 221)
m.addConstr(11*x2 <= 245)
m.addConstr(8*x2 <= 221)
m.addConstr(8*x3 <= 245)
m.addConstr(9*x3 <= 221)
m.addConstr(6*x4 <= 245)
m.addConstr(5*x4 <= 221)
m.addConstr(11*x2**2 + 8*x3**2 >= 26)
m.addConstr(2*x0 + 8*x3 >= 16)
m.addConstr(2*x0 + 6*x4 >= 18)
m.addConstr(2*x0 + 11*x2 >= 46)
m.addConstr(2*x0 + 3*x1 + 11*x2 + 8*x3 + 6*x4 >= 46)
m.addConstr(10*x0**2 + 5*x4**2 >= 27)
m.addConstr(10*x0 + 8*x2 >= 40)
m.addConstr(6*x1**2 + 5*x4**2 >= 19)
m.addConstr(8*x2 + 5*x4 >= 40)
m.addConstr(6*x1 + 9*x3 >= 42)
m.addConstr(10*x0**2 + 8*x2**2 + 9*x3**2 >= 37)
m.addConstr(10*x0 + 6*x1 + 8*x2 + 9*x3 + 5*x4 >= 37)
m.addConstr(-3*x1**2 + 10*x3**2 >= 0)
m.addConstr(3*x1**2 - 4*x2**2 >= 0)
m.addConstr(2*x0**2 + 11*x2**2 + 6*x4**2 <= 118)
m.addConstr(11*x2**2 + 8*x3**2 + 6*x4**2 <= 111)
m.addConstr(2*x0**2 + 3*x1**2 + 6*x4**2 <= 194)
m.addConstr(3*x1**2 + 11*x2**2 + 8*x3**2 <= 202)
m.addConstr(2*x0**2 + 8*x3**2 + 6*x4**2 <= 203)
m.addConstr(2*x0 + 3*x1 + 11*x2 <= 232)
m.addConstr(6*x1 + 9*x3 <= 109)
m.addConstr(6*x1 + 8*x2 + 5*x4 <= 119)
m.addConstr(10*x0 + 6*x1 + 8*x2 <= 209)
m.addConstr(10*x0**2 + 8*x2**2 + 9*x3**2 <= 153)
m.addConstr(8*x2 + 9*x3 + 5*x4 <= 156)
m.addConstr(10*x0 + 6*x1 + 5*x4 <= 160)
m.addConstr(10*x0 + 8*x2 + 5*x4 <= 182)
m.addConstr(6*x1 + 9*x3 + 5*x4 <= 204)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("packs of paper plates: ", x0.varValue)
    print("bottles of ibuprofen: ", x1.varValue)
    print("toilet paper rolls: ", x2.varValue)
    print("diapers: ", x3.varValue)
    print("cartons of milk: ", x4.varValue)
else:
    print("The model is infeasible")
```

## Step 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'packs of paper plates'), 
        ('x1', 'bottles of ibuprofen'), 
        ('x2', 'toilet paper rolls'), 
        ('x3', 'diapers'), 
        ('x4', 'cartons of milk')
    ], 
    'objective_function': '3.08*x0^2 + 5.45*x0*x1 + 3.4*x0*x2 + 4.05*x0*x3 + 5.91*x0*x4 + 2.46*x1^2 + 9.62*x1*x2 + 8.76*x1*x3 + 2.2*x2^2 + 3.71*x3^2 + 5.58*x4^2 + 5.18*x0 + 7.76*x1 + 3.37*x2 + 5.99*x3 + 8.27*x4', 
    'constraints': [
        '2*x0 <= 245', 
        '10*x0 <= 221', 
        '3*x1 <= 245', 
        '6*x1 <= 221', 
        '11*x2 <= 245', 
        '8*x2 <= 221', 
        '8*x3 <= 245', 
        '9*x3 <= 221', 
        '6*x4 <= 245', 
        '5*x4 <= 221', 
        '11*x2^2 + 8*x3^2 >= 26', 
        '2*x0 + 8*x3 >= 16', 
        '2*x0 + 6*x4 >= 18', 
        '2*x0 + 11*x2 >= 46', 
        '2*x0 + 3*x1 + 11*x2 + 8*x3 + 6*x4 >= 46', 
        '10*x0^2 + 5*x4^2 >= 27', 
        '10*x0 + 8*x2 >= 40', 
        '6*x1^2 + 5*x4^2 >= 19', 
        '8*x2 + 5*x4 >= 40', 
        '6*x1 + 9*x3 >= 42', 
        '10*x0^2 + 8*x2^2 + 9*x3^2 >= 37', 
        '10*x0 + 6*x1 + 8*x2 + 9*x3 + 5*x4 >= 37', 
        '-3*x1^2 + 10*x3^2 >= 0', 
        '3*x1^2 - 4*x2^2 >= 0', 
        '2*x0^2 + 11*x2^2 + 6*x4^2 <= 118', 
        '11*x2^2 + 8*x3^2 + 6*x4^2 <= 111', 
        '2*x0^2 + 3*x1^2 + 6*x4^2 <= 194', 
        '3*x1^2 + 11*x2^2 + 8*x3^2 <= 202', 
        '2*x0^2 + 8*x3^2 + 6*x4^2 <= 203', 
        '2*x0 + 3*x1 + 11*x2 <= 232', 
        '6*x1 + 9*x3 <= 109', 
        '6*x1 + 8*x2 + 5*x4 <= 119', 
        '10*x0 + 6*x1 + 8*x2 <= 209', 
        '10*x0^2 + 8*x2^2 + 9*x3^2 <= 153', 
        '8*x2 + 9*x3 + 5*x4 <= 156', 
        '10*x0 + 6*x1 + 5*x4 <= 160', 
        '10*x0 + 8*x2 + 5*x4 <= 182', 
        '6*x1 + 9*x3 + 5*x4 <= 204'
    ]
}
```