## Step 1: Define the optimization problem
The problem is to minimize the given objective function subject to various constraints. The objective function and constraints are defined based on the provided variables and their attributes.

## Step 2: Identify the variables and their attributes
The variables are: 
- hot dogs
- bananas
- ham sandwiches
- eggs

The attributes (resources) are given as:
- r0: grams of fiber
- r1: umami index
- r2: sourness index

With specific values for each variable:
- hot dogs: x0
- bananas: x1
- ham sandwiches: x2
- eggs: x3

## 3: Formulate the objective function
The objective function to minimize is:
\[ 5 \times (\text{hot dogs})^2 + 4 \times \text{hot dogs} \times \text{bananas} + 3 \times \text{hot dogs} \times \text{ham sandwiches} + 5 \times \text{hot dogs} \times \text{eggs} + 9 \times (\text{bananas})^2 + 1 \times \text{bananas} \times \text{ham sandwiches} + 2 \times \text{bananas} \times \text{eggs} + 5 \times (\text{ham sandwiches})^2 + 6 \times \text{ham sandwiches} \times \text{eggs} + 5 \times (\text{eggs})^2 + 6 \times \text{hot dogs} + 1 \times \text{bananas} + 5 \times \text{ham sandwiches} + 4 \times \text{eggs} \]

## 4: Define the constraints
Constraints based on the problem description:
- Fiber from hot dogs: $12 \times \text{hot dogs}$
- Umami index of hot dogs: $16 \times \text{hot dogs}$
- Sourness index of hot dogs: $13 \times \text{hot dogs}$
- Fiber from bananas: $16 \times \text{bananas}$
- Umami index of bananas: $5 \times \text{bananas}$
- Sourness index of bananas: $14 \times \text{bananas}$
- Fiber from ham sandwiches: $9 \times \text{ham sandwiches}$
- Umami index of ham sandwiches: $12 \times \text{ham sandwiches}$
- Sourness index of ham sandwiches: $8 \times \text{ham sandwiches}$
- Fiber from eggs: $17 \times \text{eggs}$
- Umami index of eggs: $12 \times \text{eggs}$
- Sourness index of eggs: $1 \times \text{eggs}$

And the specific constraints:
1. $12 \times \text{hot dogs} + 17 \times \text{eggs} \geq 39$
2. $12 \times \text{hot dogs} + 16 \times \text{bananas} \geq 21$
3. $9 \times \text{ham sandwiches} + 17 \times \text{eggs} \geq 33$
4. $16 \times \text{bananas} + 9 \times \text{ham sandwiches} + 17 \times \text{eggs} \geq 34$
5. $12 \times \text{hot dogs} + 16 \times \text{bananas} + 9 \times \text{ham sandwiches} + 17 \times \text{eggs} \geq 34$
6. $5 \times \text{bananas} + 12 \times \text{eggs} \geq 13$
7. $16 \times \text{hot dogs} + 5 \times \text{bananas} \geq 8$
8. $(16 \times \text{hot dogs})^2 + (5 \times \text{bananas})^2 + (12 \times \text{eggs})^2 \geq 22$
9. $16 \times \text{hot dogs} + 5 \times \text{bananas} + 12 \times \text{ham sandwiches} + 12 \times \text{eggs} \geq 22$
10. $8 \times \text{ham sandwiches} + 1 \times \text{eggs} \geq 53$
11. $(14 \times \text{bananas})^2 + (8 \times \text{ham sandwiches})^2 \geq 87$
12. $13 \times \text{hot dogs} + 8 \times \text{ham sandwiches} \geq 53$
13. $13 \times \text{hot dogs} + 14 \times \text{bananas} + 8 \times \text{ham sandwiches} + 1 \times \text{eggs} \geq 53$
14. $5 \times \text{ham sandwiches} - 9 \times \text{eggs} \geq 0$
15. $7 \times (\text{bananas})^2 - 9 \times (\text{eggs})^2 \geq 0$
16. $12 \times \text{hot dogs} + 16 \times \text{bananas} + 17 \times \text{eggs} \leq 133$
17. $13 \times \text{hot dogs} + 8 \times \text{ham sandwiches} \leq 87$
18. $13 \times \text{hot dogs} + 14 \times \text{bananas} \leq 192$
19. $14 \times \text{bananas} + 8 \times \text{ham sandwiches} \leq 155$
20. $8 \times \text{ham sandwiches} + 1 \times \text{eggs} \leq 208$

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

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

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

# Define the objective function
m.setObjective(5 * hot_dogs**2 + 4 * hot_dogs * bananas + 3 * hot_dogs * ham_sandwiches + 5 * hot_dogs * eggs +
               9 * bananas**2 + bananas * ham_sandwiches + 2 * bananas * eggs + 5 * ham_sandwiches**2 +
               6 * ham_sandwiches * eggs + 5 * eggs**2 + 6 * hot_dogs + bananas + 5 * ham_sandwiches + 4 * eggs,
               gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(12 * hot_dogs + 17 * eggs >= 39)
m.addConstr(12 * hot_dogs + 16 * bananas >= 21)
m.addConstr(9 * ham_sandwiches + 17 * eggs >= 33)
m.addConstr(16 * bananas + 9 * ham_sandwiches + 17 * eggs >= 34)
m.addConstr(12 * hot_dogs + 16 * bananas + 9 * ham_sandwiches + 17 * eggs >= 34)
m.addConstr(5 * bananas + 12 * eggs >= 13)
m.addConstr(16 * hot_dogs + 5 * bananas >= 8)
m.addConstr((16 * hot_dogs)**2 + (5 * bananas)**2 + (12 * eggs)**2 >= 22)
m.addConstr(16 * hot_dogs + 5 * bananas + 12 * ham_sandwiches + 12 * eggs >= 22)
m.addConstr(8 * ham_sandwiches + eggs >= 53)
m.addConstr((14 * bananas)**2 + (8 * ham_sandwiches)**2 >= 87)
m.addConstr(13 * hot_dogs + 8 * ham_sandwiches >= 53)
m.addConstr(13 * hot_dogs + 14 * bananas + 8 * ham_sandwiches + eggs >= 53)
m.addConstr(5 * ham_sandwiches - 9 * eggs >= 0)
m.addConstr(7 * bananas**2 - 9 * eggs**2 >= 0)
m.addConstr(12 * hot_dogs + 16 * bananas + 17 * eggs <= 133)
m.addConstr(13 * hot_dogs + 8 * ham_sandwiches <= 87)
m.addConstr(13 * hot_dogs + 14 * bananas <= 192)
m.addConstr(14 * bananas + 8 * ham_sandwiches <= 155)
m.addConstr(8 * ham_sandwiches + eggs <= 208)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.Status.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hot dogs: ", hot_dogs.varValue)
    print("Bananas: ", bananas.varValue)
    print("Ham sandwiches: ", ham_sandwiches.varValue)
    print("Eggs: ", eggs.varValue)
else:
    print("The model is infeasible")
```