## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'peanutbutter sandwiches' = $x_1$
- 'granola bars' = $x_2$
- 'chicken thighs' = $x_3$
- 'pickles' = $x_4$

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is: $3.35x_1^2 + 8.9x_1x_2 + 1.56x_1x_3 + 9.04x_3x_4$

## Step 3: List the constraints in symbolic notation
The constraints are:
1. $21.24x_1 + 6.28x_2 + 6.16x_3 + 4.23x_4 \leq 374$
2. $21.24x_1 \geq 0$
3. $6.28x_2 \geq 0$
4. $6.16x_3 \geq 0$
5. $4.23x_4 \geq 0$
6. $21.24^2x_1^2 + 4.23^2x_4^2 \geq 84$
7. $6.16^2x_3^2 + 4.23^2x_4^2 \geq 87$
8. $6.28x_2 + 4.23x_4 \geq 89$
9. $21.24x_1 + 6.28x_2 + 4.23x_4 \geq 57$
10. $21.24x_1 + 6.16x_3 + 4.23x_4 \geq 57$
11. $21.24x_1 + 6.28x_2 + 6.16x_3 \geq 57$
12. $21.24x_1 + 6.28x_2 + 4.23x_4 \geq 79$
13. $21.24x_1 + 6.16x_3 + 4.23x_4 \geq 79$
14. $21.24x_1 + 6.28x_2 + 6.16x_3 \geq 79$
15. $21.24x_1 + 6.28x_2 + 4.23x_4 \geq 71$
16. $21.24x_1 + 6.16x_3 + 4.23x_4 \geq 71$
17. $21.24x_1 + 6.28x_2 + 6.16x_3 \geq 71$
18. $21.24x_1 + 6.28x_2 + 6.16x_3 + 4.23x_4 \geq 71$
19. $9x_1^2 - x_2^2 \geq 0$
20. $9x_1 - 3x_3 \geq 0$
21. $6.16x_3 + 4.23x_4 \leq 108$
22. $21.24x_1 + 4.23x_4 \leq 185$
23. $21.24x_1 + 6.16x_3 + 4.23x_4 \leq 338$

## Step 4: Create the Gurobi model and variables
We will now create a Gurobi model and define the variables.

## 5: Implement the objective function and constraints in Gurobi

```python
import gurobi

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

# Define the variables
x1 = m.addVar(name="peanutbutter_sandwiches", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x2 = m.addVar(name="granola_bars", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x3 = m.addVar(name="chicken_thighs", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
x4 = m.addVar(name="pickles", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

# Objective function
m.setObjective(3.35 * x1**2 + 8.9 * x1 * x2 + 1.56 * x1 * x3 + 9.04 * x3 * x4, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(21.24 * x1 + 6.28 * x2 + 6.16 * x3 + 4.23 * x4 <= 374)
m.addConstr(21.24**2 * x1**2 + 4.23**2 * x4**2 >= 84)
m.addConstr(6.16**2 * x3**2 + 4.23**2 * x4**2 >= 87)
m.addConstr(6.28 * x2 + 4.23 * x4 >= 89)
m.addConstr(21.24 * x1 + 6.28 * x2 + 4.23 * x4 >= 57)
m.addConstr(21.24 * x1 + 6.16 * x3 + 4.23 * x4 >= 57)
m.addConstr(21.24 * x1 + 6.28 * x2 + 6.16 * x3 >= 57)
m.addConstr(21.24 * x1 + 6.28 * x2 + 4.23 * x4 >= 79)
m.addConstr(21.24 * x1 + 6.16 * x3 + 4.23 * x4 >= 79)
m.addConstr(21.24 * x1 + 6.28 * x2 + 6.16 * x3 >= 79)
m.addConstr(21.24 * x1 + 6.28 * x2 + 4.23 * x4 >= 71)
m.addConstr(21.24 * x1 + 6.16 * x3 + 4.23 * x4 >= 71)
m.addConstr(21.24 * x1 + 6.28 * x2 + 6.16 * x3 >= 71)
m.addConstr(21.24 * x1 + 6.28 * x2 + 6.16 * x3 + 4.23 * x4 >= 71)
m.addConstr(9 * x1**2 - x2**2 >= 0)
m.addConstr(9 * x1 - 3 * x3 >= 0)
m.addConstr(6.16 * x3 + 4.23 * x4 <= 108)
m.addConstr(21.24 * x1 + 4.23 * x4 <= 185)
m.addConstr(21.24 * x1 + 6.16 * x3 + 4.23 * x4 <= 338)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("peanutbutter sandwiches: ", x1.varValue)
    print("granola bars: ", x2.varValue)
    print("chicken thighs: ", x3.varValue)
    print("pickles: ", x4.varValue)
else:
    print("The model is infeasible")
```

```json
{
    'sym_variables': [
        ('x1', 'peanutbutter sandwiches'), 
        ('x2', 'granola bars'), 
        ('x3', 'chicken thighs'), 
        ('x4', 'pickles')
    ], 
    'objective_function': '3.35x1^2 + 8.9x1x2 + 1.56x1x3 + 9.04x3x4', 
    'constraints': [
        '21.24x1 + 6.28x2 + 6.16x3 + 4.23x4 <= 374', 
        '21.24^2x1^2 + 4.23^2x4^2 >= 84', 
        '6.16^2x3^2 + 4.23^2x4^2 >= 87', 
        '6.28x2 + 4.23x4 >= 89', 
        '21.24x1 + 6.28x2 + 4.23x4 >= 57', 
        '21.24x1 + 6.16x3 + 4.23x4 >= 57', 
        '21.24x1 + 6.28x2 + 6.16x3 >= 57', 
        '21.24x1 + 6.28x2 + 4.23x4 >= 79', 
        '21.24x1 + 6.16x3 + 4.23x4 >= 79', 
        '21.24x1 + 6.28x2 + 6.16x3 >= 79', 
        '21.24x1 + 6.28x2 + 4.23x4 >= 71', 
        '21.24x1 + 6.16x3 + 4.23x4 >= 71', 
        '21.24x1 + 6.28x2 + 6.16x3 >= 71', 
        '21.24x1 + 6.28x2 + 6.16x3 + 4.23x4 >= 71', 
        '9x1^2 - x2^2 >= 0', 
        '9x1 - 3x3 >= 0', 
        '6.16x3 + 4.23x4 <= 108', 
        '21.24x1 + 4.23x4 <= 185', 
        '21.24x1 + 6.16x3 + 4.23x4 <= 338'
    ]
}
```