## Step 1: Define the symbolic representation of the variables
The variables are ['potatoes', 'milkshakes', 'protein bars', 'hot dogs'], which can be represented symbolically as ['x1', 'x2', 'x3', 'x4'].

## Step 2: Create a dictionary mapping natural language objects to symbolic variables
The symbolic variables and their corresponding natural language objects are:
- x1: potatoes
- x2: milkshakes
- x3: protein bars
- x4: hot dogs

## Step 3: Formulate the objective function using symbolic variables
The objective function to minimize is: $5.19x_1^2 + 6.13x_1x_2 + 5.47x_1x_3 + 9.83x_1x_4 + 7.3x_2^2 + 4.72x_2x_3 + 3.82x_3^2 + 8.95x_4^2 + 6.59x_1 + 6.2x_4$.

## 4: Define the constraints
The constraints given are:
- $3x_1 \leq 158$ (tastiness rating of potatoes)
- $8x_2 \leq 158$ (tastiness rating of milkshakes)
- $1x_3 \leq 158$ (tastiness rating of protein bars)
- $4x_4 \leq 158$ (tastiness rating of hot dogs)
- $3^2x_1^2 + 1^2x_3^2 \geq 30$ (combined tastiness rating from potatoes and protein bars)
- $8x_2 + 4x_4 \geq 28$ (combined tastiness rating from milkshakes and hot dogs)
- $3x_1 + 8x_2 + 1x_3 + 4x_4 \geq 28$ (combined tastiness rating from all)
- $3x_1 + 1x_3 \leq 61$ (combined tastiness rating from potatoes and protein bars)
- $3x_1 + 8x_2 \leq 62$ (combined tastiness rating from potatoes and milkshakes)
- $3x_1 + 4x_4 \leq 127$ (combined tastiness rating from potatoes and hot dogs)

## 5: Convert the problem into a Gurobi code
We will use Gurobi to solve this optimization problem.

```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x1 = m.addVar(name="potatoes", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x2 = m.addVar(name="milkshakes", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x3 = m.addVar(name="protein_bars", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x4 = m.addVar(name="hot_dogs", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)

# Define the objective function
m.setObjective(5.19*x1**2 + 6.13*x1*x2 + 5.47*x1*x3 + 9.83*x1*x4 + 7.3*x2**2 + 4.72*x2*x3 + 3.82*x3**2 + 8.95*x4**2 + 6.59*x1 + 6.2*x4, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(3*x1 <= 158, name="potatoes_tastiness")
m.addConstr(8*x2 <= 158, name="milkshakes_tastiness")
m.addConstr(x3 <= 158, name="protein_bars_tastiness")
m.addConstr(4*x4 <= 158, name="hot_dogs_tastiness")
m.addConstr(9*x1**2 + x3**2 >= 30, name="potatoes_protein_bars_tastiness")
m.addConstr(8*x2 + 4*x4 >= 28, name="milkshakes_hot_dogs_tastiness")
m.addConstr(3*x1 + 8*x2 + x3 + 4*x4 >= 28, name="all_tastiness")
m.addConstr(3*x1 + x3 <= 61, name="potatoes_protein_bars_tastiness_limit")
m.addConstr(3*x1 + 8*x2 <= 62, name="potatoes_milkshakes_tastiness_limit")
m.addConstr(3*x1 + 4*x4 <= 127, name="potatoes_hot_dogs_tastiness_limit")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Potatoes: {x1.varValue}")
    print(f"Milkshakes: {x2.varValue}")
    print(f"Protein Bars: {x3.varValue}")
    print(f"Hot Dogs: {x4.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 6: Provide the symbolic representation of the problem
The symbolic representation is as follows:

```json
{
    "sym_variables": [
        ["x1", "potatoes"],
        ["x2", "milkshakes"],
        ["x3", "protein bars"],
        ["x4", "hot dogs"]
    ],
    "objective_function": "5.19x1^2 + 6.13x1x2 + 5.47x1x3 + 9.83x1x4 + 7.3x2^2 + 4.72x2x3 + 3.82x3^2 + 8.95x4^2 + 6.59x1 + 6.2x4",
    "constraints": [
        "3x1 <= 158",
        "8x2 <= 158",
        "x3 <= 158",
        "4x4 <= 158",
        "9x1^2 + x3^2 >= 30",
        "8x2 + 4x4 >= 28",
        "3x1 + 8x2 + x3 + 4x4 >= 28",
        "3x1 + x3 <= 61",
        "3x1 + 8x2 <= 62",
        "3x1 + 4x4 <= 127"
    ]
}
```