## Step 1: Define the symbolic representation of the problem
The problem involves variables: eggs, bananas, protein bars, and hot dogs, which we can denote as $x_0, x_1, x_2, x_3$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $5x_0^2 + 3x_0x_2 + 6x_1^2 + x_1x_2 + 6x_1x_3 + 5x_2^2 + 2x_0 + 2x_3$.

## 3: List the constraints in symbolic notation
Constraints:
1. $20x_0 + 3x_1 + 5x_2 + 4x_3 \leq 264$
2. $15x_0 + 6x_1 + 6x_2 + 26x_3 \leq 163$
3. $12x_0 + 14x_1 + 19x_2 + 20x_3 \leq 239$
4. $3x_1^2 + 4x_3^2 \geq 55$
5. $3x_1^2 + 5x_2^2 \geq 31$
6. $20x_0^2 + 4x_3^2 \geq 48$
7. $20x_0 + 3x_1 + 5x_2 + 4x_3 \geq 48$
8. $6x_1^2 + 26x_3^2 \geq 15$
9. $15x_0 + 6x_2 \geq 17$
10. $15x_0 + 6x_1 + 6x_2 + 26x_3 \geq 17$
11. $12x_0 + 20x_3 \geq 48$
12. $12x_0 + 19x_2 \geq 46$
13. $12x_0 + 14x_1 \geq 50$
14. $14x_1 + 19x_2 \geq 54$
15. $12x_0^2 + 14x_1^2 + 19x_2^2 \geq 30$
16. $12x_0 + 14x_1 + 20x_3 \geq 30$
17. $12x_0 + 19x_2 + 20x_3 \geq 30$
18. $14x_1 + 19x_2 + 20x_3 \geq 30$
19. $12x_0^2 + 14x_1^2 + 19x_2^2 \geq 46$
20. $12x_0^2 + 14x_1^2 + 20x_3^2 \geq 46$
21. $12x_0 + 19x_2 + 20x_3 \geq 46$
22. $14x_1 + 19x_2 + 20x_3 \geq 46$
23. $12x_0 + 14x_1 + 19x_2 \geq 56$
24. $12x_0^2 + 14x_1^2 + 20x_3^2 \geq 56$
25. $12x_0 + 19x_2 + 20x_3 \geq 56$
26. $14x_1 + 19x_2 + 20x_3 \geq 56$
27. $12x_0 + 14x_1 + 19x_2 \geq 51$
28. $12x_0 + 14x_1 + 20x_3 \geq 51$
29. $12x_0 + 19x_2 + 20x_3 \geq 51$
30. $14x_1^2 + 19x_2^2 + 20x_3^2 \geq 51$
31. $12x_0 + 14x_1 + 19x_2 + 20x_3 \geq 51$
32. $-6x_2^2 + 7x_3^2 \geq 0$
33. $x_1 - 2x_3 \geq 0$
34. $15^2x_0^2 + 6^2x_1^2 \leq 82$
35. $6x_2 + 26x_3 \leq 163$
36. $15x_0 + 26x_3 \leq 76$
37. $6x_1 + 26x_3 \leq 163$
38. $15x_0 + 6x_2 \leq 59$
39. $12x_0 + 14x_1 \leq 76$
40. $12x_0 + 19x_2 \leq 234$
41. $12x_0^2 + 19x_2^2 + 20x_3^2 \leq 164$

## 4: Define the symbolic variables and constraints for the JSON output
```json
{
    "sym_variables": [
        ["x0", "eggs"],
        ["x1", "bananas"],
        ["x2", "protein bars"],
        ["x3", "hot dogs"]
    ],
    "objective_function": "5*x0^2 + 3*x0*x2 + 6*x1^2 + x1*x2 + 6*x1*x3 + 5*x2^2 + 2*x0 + 2*x3",
    "constraints": [
        "20*x0 + 3*x1 + 5*x2 + 4*x3 <= 264",
        "15*x0 + 6*x1 + 6*x2 + 26*x3 <= 163",
        "12*x0 + 14*x1 + 19*x2 + 20*x3 <= 239",
        "3*x1^2 + 4*x3^2 >= 55",
        "3*x1^2 + 5*x2^2 >= 31",
        "20*x0^2 + 4*x3^2 >= 48",
        "20*x0 + 3*x1 + 5*x2 + 4*x3 >= 48",
        "6*x1^2 + 26*x3^2 >= 15",
        "15*x0 + 6*x2 >= 17",
        "15*x0 + 6*x1 + 6*x2 + 26*x3 >= 17",
        "12*x0 + 20*x3 >= 48",
        "12*x0 + 19*x2 >= 46",
        "12*x0 + 14*x1 >= 50",
        "14*x1 + 19*x2 >= 54",
        "12*x0^2 + 14*x1^2 + 19*x2^2 >= 30",
        "12*x0 + 14*x1 + 20*x3 >= 30",
        "12*x0 + 19*x2 + 20*x3 >= 30",
        "14*x1 + 19*x2 + 20*x3 >= 30",
        "12*x0^2 + 14*x1^2 + 19*x2^2 >= 46",
        "12*x0^2 + 14*x1^2 + 20*x3^2 >= 46",
        "12*x0 + 19*x2 + 20*x3 >= 46",
        "14*x1 + 19*x2 + 20*x3 >= 46",
        "12*x0 + 14*x1 + 19*x2 >= 56",
        "12*x0^2 + 14*x1^2 + 20*x3^2 >= 56",
        "12*x0 + 19*x2 + 20*x3 >= 56",
        "14*x1 + 19*x2 + 20*x3 >= 56",
        "12*x0 + 14*x1 + 19*x2 >= 51",
        "12*x0 + 14*x1 + 20*x3 >= 51",
        "12*x0 + 19*x2 + 20*x3 >= 51",
        "14*x1^2 + 19*x2^2 + 20*x3^2 >= 51",
        "12*x0 + 14*x1 + 19*x2 + 20*x3 >= 51",
        "-6*x2^2 + 7*x3^2 >= 0",
        "x1 - 2*x3 >= 0",
        "225*x0^2 + 36*x1^2 <= 82",
        "6*x2 + 26*x3 <= 163",
        "15*x0 + 26*x3 <= 76",
        "6*x1 + 26*x3 <= 163",
        "15*x0 + 6*x2 <= 59",
        "12*x0 + 14*x1 <= 76",
        "12*x0 + 19*x2 <= 234",
        "12*x0^2 + 19*x2^2 + 20*x3^2 <= 164"
    ]
}
```

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

# Define the model
model = gurobi.Model()

# Define the variables
x0 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="eggs")
x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="bananas")
x2 = model.addVar(lb=0, ub=gurobi.GRB.INFINITY, name="protein bars", vtype=gurobi.GRB.INTEGER)
x3 = model.addVar(lb=0, ub=gurobi.GRB.INFINITY, name="hot dogs", vtype=gurobi.GRB.INTEGER)

# Define the objective function
model.setObjective(5*x0**2 + 3*x0*x2 + 6*x1**2 + x1*x2 + 6*x1*x3 + 5*x2**2 + 2*x0 + 2*x3, gurobi.GRB.MINIMIZE)

# Add constraints
model.addConstr(20*x0 + 3*x1 + 5*x2 + 4*x3 <= 264)
model.addConstr(15*x0 + 6*x1 + 6*x2 + 26*x3 <= 163)
model.addConstr(12*x0 + 14*x1 + 19*x2 + 20*x3 <= 239)
model.addConstr(3*x1**2 + 4*x3**2 >= 55)
model.addConstr(3*x1**2 + 5*x2**2 >= 31)
model.addConstr(20*x0**2 + 4*x3**2 >= 48)
model.addConstr(20*x0 + 3*x1 + 5*x2 + 4*x3 >= 48)
model.addConstr(6*x1**2 + 26*x3**2 >= 15)
model.addConstr(15*x0 + 6*x2 >= 17)
model.addConstr(15*x0 + 6*x1 + 6*x2 + 26*x3 >= 17)
model.addConstr(12*x0 + 20*x3 >= 48)
model.addConstr(12*x0 + 19*x2 >= 46)
model.addConstr(12*x0 + 14*x1 >= 50)
model.addConstr(14*x1 + 19*x2 >= 54)
model.addConstr(12*x0**2 + 14*x1**2 + 19*x2**2 >= 30)
model.addConstr(12*x0 + 14*x1 + 20*x3 >= 30)
model.addConstr(12*x0 + 19*x2 + 20*x3 >= 30)
model.addConstr(14*x1 + 19*x2 + 20*x3 >= 30)
model.addConstr(12*x0**2 + 14*x1**2 + 19*x2**2 >= 46)
model.addConstr(12*x0**2 + 14*x1**2 + 20*x3**2 >= 46)
model.addConstr(12*x0 + 19*x2 + 20*x3 >= 46)
model.addConstr(14*x1 + 19*x2 + 20*x3 >= 46)
model.addConstr(12*x0 + 14*x1 + 19*x2 >= 56)
model.addConstr(12*x0**2 + 14*x1**2 + 20*x3**2 >= 56)
model.addConstr(12*x0 + 19*x2 + 20*x3 >= 56)
model.addConstr(14*x1 + 19*x2 + 20*x3 >= 56)
model.addConstr(12*x0 + 14*x1 + 19*x2 >= 51)
model.addConstr(12*x0 + 14*x1 + 20*x3 >= 51)
model.addConstr(12*x0 + 19*x2 + 20*x3 >= 51)
model.addConstr(14*x1**2 + 19*x2**2 + 20*x3**2 >= 51)
model.addConstr(12*x0 + 14*x1 + 19*x2 + 20*x3 >= 51)
model.addConstr(-6*x2**2 + 7*x3**2 >= 0)
model.addConstr(x1 - 2*x3 >= 0)
model.addConstr(225*x0**2 + 36*x1**2 <= 82)
model.addConstr(6*x2 + 26*x3 <= 163)
model.addConstr(15*x0 + 26*x3 <= 76)
model.addConstr(6*x1 + 26*x3 <= 163)
model.addConstr(15*x0 + 6*x2 <= 59)
model.addConstr(12*x0 + 14*x1 <= 76)
model.addConstr(12*x0 + 19*x2 <= 234)
model.addConstr(12*x0**2 + 19*x2**2 + 20*x3**2 <= 164)

# Solve the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objval)
    print("Eggs: ", x0.varValue)
    print("Bananas: ", x1.varValue)
    print("Protein Bars: ", x2.varValue)
    print("Hot Dogs: ", x3.varValue)
else:
    print("No solution found")
```