## Step 1: Define the symbolic representation of the variables
The variables are 'hot dogs', 'bagged salads', and 'peanutbutter sandwiches', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $4x_0x_1 + 8x_0x_2 + 4x_1^2 + 7x_1x_2 + 5x_0 + 2x_1 + 3x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $x_0 \geq 0$, $x_1 \geq 0$, $x_2 \geq 0$ (non-negativity constraints, but not explicitly mentioned, so we assume they are)
- $x_0 \cdot 1 \leq 59$ (iron from hot dogs)
- $x_0 \cdot 3 \leq 86$ (fiber from hot dogs)
- $x_1 \cdot 5 \leq 59$ (iron from bagged salads)
- $x_1 \cdot 5 \leq 86$ (fiber from bagged salads)
- $x_2 \cdot 8 \leq 59$ (iron from peanutbutter sandwiches)
- $x_2 \cdot 3 \leq 86$ (fiber from peanutbutter sandwiches)
- $x_0^2 + x_2^2 \geq 16$ (iron from hot dogs and peanutbutter sandwiches squared)
- $x_1 + x_2 \geq 9$ (iron from bagged salads and peanutbutter sandwiches)
- $x_0^2 + x_1^2 + x_2^2 \geq 14$ (iron from hot dogs, bagged salads, and peanutbutter sandwiches squared)
- $x_0 + x_1 + x_2 \geq 14$ (iron from hot dogs, bagged salads, and peanutbutter sandwiches)
- $x_1 + x_2 \geq 27$ (fiber from bagged salads and peanutbutter sandwiches)
- $x_0 + x_1 + x_2 \geq 27$ (fiber from hot dogs, bagged salads, and peanutbutter sandwiches)
- $x_0 - 7x_1 \geq 0$
- $5x_0 - 8x_2 \geq 0$
- $x_1 + x_2 \leq 21$ (iron from bagged salads and peanutbutter sandwiches)
- $x_0^2 + x_2^2 \leq 39$ (iron from hot dogs and peanutbutter sandwiches squared)

## 4: Create a symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'hot dogs'), ('x1', 'bagged salads'), ('x2', 'peanutbutter sandwiches')],
    'objective_function': '4*x0*x1 + 8*x0*x2 + 4*x1^2 + 7*x1*x2 + 5*x0 + 2*x1 + 3*x2',
    'constraints': [
        'x0 >= 0', 'x1 >= 0', 'x2 >= 0',
        'x0 <= 59', '3*x0 <= 86', 
        '5*x1 <= 59', '5*x1 <= 86',
        '8*x2 <= 59', '3*x2 <= 86',
        'x0^2 + x2^2 >= 16', 'x1 + x2 >= 9',
        'x0^2 + x1^2 + x2^2 >= 14', 'x0 + x1 + x2 >= 14',
        'x1 + x2 >= 27', 'x0 + x1 + x2 >= 27',
        'x0 - 7*x1 >= 0', '5*x0 - 8*x2 >= 0',
        'x1 + x2 <= 21', 'x0^2 + x2^2 <= 39'
    ]
}
```

## 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="bagged_salads", lb=0)
x2 = m.addVar(name="peanutbutter_sandwiches", lb=0)

# Objective function
m.setObjective(4*x0*x1 + 8*x0*x2 + 4*x1**2 + 7*x1*x2 + 5*x0 + 2*x1 + 3*x2, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(x0 <= 59)
m.addConstr(3*x0 <= 86)
m.addConstr(5*x1 <= 59)
m.addConstr(5*x1 <= 86)
m.addConstr(8*x2 <= 59)
m.addConstr(3*x2 <= 86)
m.addConstr(x0**2 + x2**2 >= 16)
m.addConstr(x1 + x2 >= 9)
m.addConstr(x0**2 + x1**2 + x2**2 >= 14)
m.addConstr(x0 + x1 + x2 >= 14)
m.addConstr(x1 + x2 >= 27)
m.addConstr(x0 + x1 + x2 >= 27)
m.addConstr(x0 - 7*x1 >= 0)
m.addConstr(5*x0 - 8*x2 >= 0)
m.addConstr(x1 + x2 <= 21)
m.addConstr(x0**2 + x2**2 <= 39)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hot dogs: ", x0.varValue)
    print("Bagged salads: ", x1.varValue)
    print("Peanutbutter sandwiches: ", x2.varValue)
else:
    print("The model is infeasible")
```