## Step 1: Define the symbolic representation of the variables
The variables are 'apple pies', 'chicken thighs', 'hamburgers', and 'eggs', which can be represented symbolically as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $5.22x_0x_1 + 7.67x_0x_3 + 5.72x_1x_2 + 3.72x_1x_3 + 8.75x_0 + 4.07x_3$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $5.76x_1 + 4.74x_2 \geq 17$ (healthiness rating of chicken thighs and hamburgers)
- $4.74^2x_2^2 + 3.35^2x_3^2 \geq 26$ (healthiness rating of hamburgers and eggs)
- $2.59x_0 + 5.76x_1 + 4.74x_2 + 3.35x_3 \geq 26$ (total healthiness rating)
- $9.41x_0 + 6.55x_1 \geq 41$ (protein from apple pies and chicken thighs)
- $9.41x_0 + 0.76x_2 \geq 53$ (protein from apple pies and hamburgers)
- $9.41x_0 + 4.79x_3 \geq 51$ (protein from apple pies and eggs)
- $0.76x_2 + 4.79x_3 \geq 42$ (protein from hamburgers and eggs)
- $6.55x_1 + 0.76x_2 \geq 27$ (protein from chicken thighs and hamburgers)
- $6.55^2x_1^2 + 4.79^2x_3^2 \geq 58$ (protein from chicken thighs and eggs)
- $9.41^2x_0^2 + 6.55^2x_1^2 + 4.79^2x_3^2 \geq 32$ (protein from apple pies, chicken thighs, and eggs)
- $9.41x_0 + 6.55x_1 + 0.76x_2 + 4.79x_3 \geq 32$ (total protein)
- $-3x_0 + 6x_3 \geq 0$ (relationship between apple pies and eggs)
- $-10x_0 + 6x_1 \geq 0$ (relationship between apple pies and chicken thighs)
- $4.74x_2 + 3.35x_3 \leq 33$ (healthiness rating of hamburgers and eggs)
- $2.59x_0 + 5.76x_1 + 4.74x_2 \leq 61$ (healthiness rating of apple pies, chicken thighs, and hamburgers)
- $2.59x_0 + 5.76x_1 + 3.35x_3 \leq 48$ (healthiness rating of apple pies, chicken thighs, and eggs)
- $5.76x_1 + 4.74x_2 + 3.35x_3 \leq 50$ (healthiness rating of chicken thighs, hamburgers, and eggs)
- $9.41x_0 + 4.79x_3 \leq 195$ (protein from apple pies and eggs)

## 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'apple pies'),
        ('x1', 'chicken thighs'),
        ('x2', 'hamburgers'),
        ('x3', 'eggs')
    ],
    'objective_function': '5.22*x0*x1 + 7.67*x0*x3 + 5.72*x1*x2 + 3.72*x1*x3 + 8.75*x0 + 4.07*x3',
    'constraints': [
        '5.76*x1 + 4.74*x2 >= 17',
        '4.74^2*x2^2 + 3.35^2*x3^2 >= 26',
        '2.59*x0 + 5.76*x1 + 4.74*x2 + 3.35*x3 >= 26',
        '9.41*x0 + 6.55*x1 >= 41',
        '9.41*x0 + 0.76*x2 >= 53',
        '9.41*x0 + 4.79*x3 >= 51',
        '0.76*x2 + 4.79*x3 >= 42',
        '6.55*x1 + 0.76*x2 >= 27',
        '6.55^2*x1^2 + 4.79^2*x3^2 >= 58',
        '9.41^2*x0^2 + 6.55^2*x1^2 + 4.79^2*x3^2 >= 32',
        '9.41*x0 + 6.55*x1 + 0.76*x2 + 4.79*x3 >= 32',
        '-3*x0 + 6*x3 >= 0',
        '-10*x0 + 6*x1 >= 0',
        '4.74*x2 + 3.35*x3 <= 33',
        '2.59*x0 + 5.76*x1 + 4.74*x2 <= 61',
        '2.59*x0 + 5.76*x1 + 3.35*x3 <= 48',
        '5.76*x1 + 4.74*x2 + 3.35*x3 <= 50',
        '9.41*x0 + 4.79*x3 <= 195'
    ]
}
```

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

# Define the model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name='apple_pies', lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x1 = m.addVar(name='chicken_thighs', lb=0, ub=gurobi.GRB.INFINITY, integrality=1)
x2 = m.addVar(name='hamburgers', lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x3 = m.addVar(name='eggs', lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

# Define the objective function
m.setObjective(5.22*x0*x1 + 7.67*x0*x3 + 5.72*x1*x2 + 3.72*x1*x3 + 8.75*x0 + 4.07*x3, gurobi.GRB.MINIMIZE)

# Define the constraints
m.addConstr(5.76*x1 + 4.74*x2 >= 17)
m.addConstr(4.74**2*x2**2 + 3.35**2*x3**2 >= 26)
m.addConstr(2.59*x0 + 5.76*x1 + 4.74*x2 + 3.35*x3 >= 26)
m.addConstr(9.41*x0 + 6.55*x1 >= 41)
m.addConstr(9.41*x0 + 0.76*x2 >= 53)
m.addConstr(9.41*x0 + 4.79*x3 >= 51)
m.addConstr(0.76*x2 + 4.79*x3 >= 42)
m.addConstr(6.55*x1 + 0.76*x2 >= 27)
m.addConstr(6.55**2*x1**2 + 4.79**2*x3**2 >= 58)
m.addConstr(9.41**2*x0**2 + 6.55**2*x1**2 + 4.79**2*x3**2 >= 32)
m.addConstr(9.41*x0 + 6.55*x1 + 0.76*x2 + 4.79*x3 >= 32)
m.addConstr(-3*x0 + 6*x3 >= 0)
m.addConstr(-10*x0 + 6*x1 >= 0)
m.addConstr(4.74*x2 + 3.35*x3 <= 33)
m.addConstr(2.59*x0 + 5.76*x1 + 4.74*x2 <= 61)
m.addConstr(2.59*x0 + 5.76*x1 + 3.35*x3 <= 48)
m.addConstr(5.76*x1 + 4.74*x2 + 3.35*x3 <= 50)
m.addConstr(9.41*x0 + 4.79*x3 <= 195)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.Status.OPTIMAL:
    print('Objective: ', m.objVal)
    print('Apple pies: ', x0.varValue)
    print('Chicken thighs: ', x1.varValue)
    print('Hamburgers: ', x2.varValue)
    print('Eggs: ', x3.varValue)
else:
    print('No solution found')
```