## Step 1: Define the variables and their attributes
We have the following variables: ['chicken thighs', 'bagged salads', 'hamburgers', 'granola bars', 'corn cobs']. Let's denote them as $x_0, x_1, x_2, x_3, x_4$ respectively.

## Step 2: Define the objective function
The objective function to minimize is: $8.94x_0x_1 + 3.49x_0x_3 + 6.96x_1^2 + 1.22x_1x_4 + 5.04x_2x_3 + 9.86x_2 + 1.62x_4$.

## 3: Define the constraints
The constraints are:
- $5x_0 + 5x_1 + x_2 + 2x_3 + 2x_4 \leq 214$ (carbohydrates upper bound)
- $5x_0 \geq 0$ (carbohydrates in chicken thighs)
- $5x_1 \geq 0$ (carbohydrates in bagged salads)
- $x_2 \geq 0$ (carbohydrates in hamburgers)
- $2x_3 \geq 0$ (carbohydrates in granola bars)
- $2x_4 \geq 0$ (carbohydrates in corn cobs)
- $x_2^2 + x_3^2 \geq 31$ (carbohydrates from hamburgers and granola bars)
- $5x_0 + 5x_1 \geq 24$ (carbohydrates from chicken thighs and bagged salads)
- $5x_0 + 2x_3 \geq 24$ (carbohydrates from chicken thighs and granola bars)
- $5x_1 + 2x_4 \geq 23$ (carbohydrates from bagged salads and corn cobs)
- $x_2 + 2x_4 \geq 17$ (carbohydrates from hamburgers and corn cobs)
- $5x_1 + 2x_3 \geq 39$ (carbohydrates from bagged salads and granola bars)
- $5x_0 + x_2 + 2x_4 \geq 21$ (carbohydrates from chicken thighs, hamburgers, and corn cobs)
- $5x_0^2 + 5x_1^2 + 2x_4^2 \geq 21$ (carbohydrates from chicken thighs, bagged salads, and corn cobs squared)
- $x_2 + 2x_3 + 2x_4 \geq 21$ (carbohydrates from hamburgers, granola bars, and corn cobs)
- $5x_0 + x_2 + 2x_4 \geq 29$ (carbohydrates from chicken thighs, hamburgers, and corn cobs)
- $5x_0 + 5x_1 + 2x_4 \geq 29$ (carbohydrates from chicken thighs, bagged salads, and corn cobs)
- $x_2 + 2x_3 + 2x_4 \geq 29$ (carbohydrates from hamburgers, granola bars, and corn cobs)
- $5x_0^2 + x_2^2 + 2x_4^2 \geq 27$ (carbohydrates from chicken thighs, hamburgers, and corn cobs squared)
- $5x_0 + 5x_1 + 2x_4 \geq 27$ (carbohydrates from chicken thighs, bagged salads, and corn cobs)
- $x_2^2 + 2x_3^2 + 2x_4^2 \geq 27$ (carbohydrates from hamburgers, granola bars, and corn cobs squared)
- $5x_0 + 5x_1 + x_2 + 2x_3 + 2x_4 \geq 27$ (carbohydrates from all)
- $-6x_1^2 + 5x_2^2 \geq 0$
- $-5x_2^2 + 8x_4^2 \geq 0$
- $5x_1^2 + x_2^2 \leq 204$
- $x_2 + 2x_3 \leq 156$
- $5x_0 + 2x_4 \leq 117$
- $5x_1 + 2x_4 \leq 81$
- $5x_0 + 2x_3 \leq 58$
- $5x_0 + x_2 \leq 54$
- $5x_0 + 5x_1 \leq 86$
- $5x_0 + 5x_1 + 2x_4 \leq 119$
- $x_2 + 2x_3 + 2x_4 \leq 93$
- $5x_1^2 + x_2^2 + 2x_4^2 \leq 103$
- $5x_0^2 + 5x_1^2 + x_2^2 \leq 194$

## 4: Implement the problem in Gurobi
```python
import gurobi

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

# Define the variables
x0 = m.addVar(name="chicken_thighs", lb=0)
x1 = m.addVar(name="bagged_salads", lb=0)
x2 = m.addVar(name="hamburgers", lb=0)
x3 = m.addVar(name="granola_bars", lb=0)
x4 = m.addVar(name="corn_cobs", lb=0)

# Objective function
m.setObjective(8.94*x0*x1 + 3.49*x0*x3 + 6.96*x1**2 + 1.22*x1*x4 + 5.04*x2*x3 + 9.86*x2 + 1.62*x4, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(5*x0 + 5*x1 + x2 + 2*x3 + 2*x4 <= 214)
m.addConstr(x2**2 + x3**2 >= 31)
m.addConstr(5*x0 + 5*x1 >= 24)
m.addConstr(5*x0 + 2*x3 >= 24)
m.addConstr(5*x1 + 2*x4 >= 23)
m.addConstr(x2 + 2*x4 >= 17)
m.addConstr(5*x1 + 2*x3 >= 39)
m.addConstr(5*x0 + x2 + 2*x4 >= 21)
m.addConstr(5*x0**2 + 5*x1**2 + 2*x4**2 >= 21)
m.addConstr(x2 + 2*x3 + 2*x4 >= 21)
m.addConstr(5*x0 + x2 + 2*x4 >= 29)
m.addConstr(5*x0 + 5*x1 + 2*x4 >= 29)
m.addConstr(x2 + 2*x3 + 2*x4 >= 29)
m.addConstr(5*x0**2 + x2**2 + 2*x4**2 >= 27)
m.addConstr(5*x0 + 5*x1 + 2*x4 >= 27)
m.addConstr(x2**2 + 2*x3**2 + 2*x4**2 >= 27)
m.addConstr(5*x0 + 5*x1 + x2 + 2*x3 + 2*x4 >= 27)
m.addConstr(-6*x1**2 + 5*x2**2 >= 0)
m.addConstr(-5*x2**2 + 8*x4**2 >= 0)
m.addConstr(5*x1**2 + x2**2 <= 204)
m.addConstr(x2 + 2*x3 <= 156)
m.addConstr(5*x0 + 2*x4 <= 117)
m.addConstr(5*x1 + 2*x4 <= 81)
m.addConstr(5*x0 + 2*x3 <= 58)
m.addConstr(5*x0 + x2 <= 54)
m.addConstr(5*x0 + 5*x1 <= 86)
m.addConstr(5*x0 + 5*x1 + 2*x4 <= 119)
m.addConstr(x2 + 2*x3 + 2*x4 <= 93)
m.addConstr(5*x1**2 + x2**2 + 2*x4**2 <= 103)
m.addConstr(5*x0**2 + 5*x1**2 + x2**2 <= 194)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Chicken thighs: ", x0.x)
    print("Bagged salads: ", x1.x)
    print("Hamburgers: ", x2.x)
    print("Granola bars: ", x3.x)
    print("Corn cobs: ", x4.x)
else:
    print("No solution found")
```