## Step 1: Define the symbolic representation of the variables
The variables are 'hot dogs', 'bananas', 'ham sandwiches', 'eggs' which can be represented symbolically as $x_0, x_1, x_2, x_3$ respectively.

## Step 2: Convert the objective function into symbolic notation
The objective function to minimize is $5x_0^2 + 4x_0x_1 + 3x_0x_2 + 5x_0x_3 + 9x_1^2 + x_1x_2 + 2x_1x_3 + 5x_2^2 + 6x_2x_3 + 5x_3^2 + 6x_0 + x_1 + 5x_2 + 4x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $12x_0 + 16x_1 + 9x_2 + 17x_3 \leq 216$ (total fiber)
- $16x_0 + 5x_1 + 12x_2 + 12x_3 \leq 93$ (total umami index)
- $13x_0 + 14x_1 + 8x_2 + x_3 \leq 349$ (total sourness index)
- $12x_0 + 17x_3 \geq 39$ (fiber from hot dogs and eggs)
- $12x_0 + 16x_1 \geq 21$ (fiber from hot dogs and bananas)
- $9x_2 + 17x_3 \geq 33$ (fiber from ham sandwiches and eggs)
- $16x_1 + 9x_2 + 17x_3 \geq 34$ (fiber from bananas, ham sandwiches, and eggs)
- $12x_0 + 16x_1 + 9x_2 + 17x_3 \geq 34$ (fiber from all)
- $5x_1 + 12x_3 \geq 13$ (umami index from bananas and eggs)
- $16x_0 + 5x_1 \geq 8$ (umami index from hot dogs and bananas)
- $(16x_0)^2 + (5x_1)^2 + (12x_3)^2 \geq 22$ (umami index from hot dogs, bananas, and eggs squared)
- $16x_0 + 5x_1 + 12x_2 + 12x_3 \geq 22$ (total umami index)
- $8x_2 + x_3 \geq 53$ (sourness index from ham sandwiches and eggs)
- $(14x_1)^2 + (8x_2)^2 \geq 87$ (sourness index from bananas and ham sandwiches squared)
- $13x_0 + 8x_2 \geq 53$ (sourness index from hot dogs and ham sandwiches)
- $13x_0 + 14x_1 + 8x_2 + x_3 \geq 53$ (total sourness index)
- $5x_2 - 9x_3 \geq 0$
- $7x_1^2 - 9x_3^2 \geq 0$
- $12x_0 + 16x_1 + 17x_3 \leq 133$ (fiber from hot dogs, bananas, and eggs)
- $13x_0 + 8x_2 \leq 87$ (sourness index from hot dogs and ham sandwiches)
- $13x_0 + 14x_1 \leq 192$ (sourness index from hot dogs and bananas)
- $14x_1 + 8x_2 \leq 155$ (sourness index from bananas and ham sandwiches)
- $8x_2 + x_3 \leq 208$ (sourness index from ham sandwiches and eggs)

## 4: Create the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hot dogs'), 
        ('x1', 'bananas'), 
        ('x2', 'ham sandwiches'), 
        ('x3', 'eggs')
    ], 
    'objective_function': '5*x0^2 + 4*x0*x1 + 3*x0*x2 + 5*x0*x3 + 9*x1^2 + x1*x2 + 2*x1*x3 + 5*x2^2 + 6*x2*x3 + 5*x3^2 + 6*x0 + x1 + 5*x2 + 4*x3', 
    'constraints': [
        '12*x0 + 16*x1 + 9*x2 + 17*x3 <= 216',
        '16*x0 + 5*x1 + 12*x2 + 12*x3 <= 93',
        '13*x0 + 14*x1 + 8*x2 + x3 <= 349',
        '12*x0 + 17*x3 >= 39',
        '12*x0 + 16*x1 >= 21',
        '9*x2 + 17*x3 >= 33',
        '16*x1 + 9*x2 + 17*x3 >= 34',
        '12*x0 + 16*x1 + 9*x2 + 17*x3 >= 34',
        '5*x1 + 12*x3 >= 13',
        '16*x0 + 5*x1 >= 8',
        '(16*x0)^2 + (5*x1)^2 + (12*x3)^2 >= 22',
        '16*x0 + 5*x1 + 12*x2 + 12*x3 >= 22',
        '8*x2 + x3 >= 53',
        '(14*x1)^2 + (8*x2)^2 >= 87',
        '13*x0 + 8*x2 >= 53',
        '13*x0 + 14*x1 + 8*x2 + x3 >= 53',
        '5*x2 - 9*x3 >= 0',
        '7*x1^2 - 9*x3^2 >= 0',
        '12*x0 + 16*x1 + 17*x3 <= 133',
        '13*x0 + 8*x2 <= 87',
        '13*x0 + 14*x1 <= 192',
        '14*x1 + 8*x2 <= 155',
        '8*x2 + x3 <= 208'
    ]
}
```

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

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

# Define the variables
x0 = m.addVar(name="hot_dogs", lb=0)
x1 = m.addVar(name="bananas", lb=0)
x2 = m.addVar(name="ham_sandwiches", lb=0)
x3 = m.addVar(name="eggs", lb=0)

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

# Add constraints
m.addConstr(12*x0 + 16*x1 + 9*x2 + 17*x3 <= 216)
m.addConstr(16*x0 + 5*x1 + 12*x2 + 12*x3 <= 93)
m.addConstr(13*x0 + 14*x1 + 8*x2 + x3 <= 349)
m.addConstr(12*x0 + 17*x3 >= 39)
m.addConstr(12*x0 + 16*x1 >= 21)
m.addConstr(9*x2 + 17*x3 >= 33)
m.addConstr(16*x1 + 9*x2 + 17*x3 >= 34)
m.addConstr(12*x0 + 16*x1 + 9*x2 + 17*x3 >= 34)
m.addConstr(5*x1 + 12*x3 >= 13)
m.addConstr(16*x0 + 5*x1 >= 8)
m.addConstr((16*x0)**2 + (5*x1)**2 + (12*x3)**2 >= 22)
m.addConstr(16*x0 + 5*x1 + 12*x2 + 12*x3 >= 22)
m.addConstr(8*x2 + x3 >= 53)
m.addConstr((14*x1)**2 + (8*x2)**2 >= 87)
m.addConstr(13*x0 + 8*x2 >= 53)
m.addConstr(13*x0 + 14*x1 + 8*x2 + x3 >= 53)
m.addConstr(5*x2 - 9*x3 >= 0)
m.addConstr(7*x1**2 - 9*x3**2 >= 0)
m.addConstr(12*x0 + 16*x1 + 17*x3 <= 133)
m.addConstr(13*x0 + 8*x2 <= 87)
m.addConstr(13*x0 + 14*x1 <= 192)
m.addConstr(14*x1 + 8*x2 <= 155)
m.addConstr(8*x2 + x3 <= 208)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("hot dogs: ", x0.varValue)
    print("bananas: ", x1.varValue)
    print("ham sandwiches: ", x2.varValue)
    print("eggs: ", x3.varValue)
else:
    print("The model is infeasible")
```