## Step 1: Define the variables and their corresponding symbolic notation
Let's denote the variables as follows:
- $x_1$ : cherry pies
- $x_2$ : knishes
- $x_3$ : chicken thighs
- $x_4$ : fruit salads
- $x_5$ : bowls of instant ramen
- $x_6$ : hot dogs

## Step 2: Formulate the objective function using the given variables
The objective function to minimize is:
\[ 4x_1^2 + 4x_1x_2 + 9x_2^2 + 8x_2x_4 + 3x_2x_5 + 8x_2x_6 + 9x_3^2 + x_3x_4 + 6x_4x_5 + 2x_6^2 + 4x_1 + 5x_2 + 4x_3 + 8x_6 \]

## 3: List all the constraints
Constraints:
1. $5x_1 + 5x_2 + 3x_3 + 5x_4 + 4x_5 + 4x_6 \leq 51$
2. $5x_2 + 4x_6 \geq 3$
3. $5x_1 + 5x_2 \geq 2$
4. $5x_1 + 4x_5 + 4x_6 \geq 7$
5. $5x_2 + 3x_3 + 5x_4 \geq 7$
6. $5x_1 + 5x_2 + 3x_3 \geq 7$
7. $5x_1^2 + 5x_2^2 + 4x_6^2 \geq 7$
8. $5x_1 + 5x_2 + 4x_5 \geq 7$
9. $3x_3 + 5x_4 + 4x_5 \geq 7$
10. $5x_1 + 5x_4 + 4x_6 \geq 7$
11. $5x_4^2 + 4x_5^2 + 4x_6^2 \geq 7$
12. $5x_2^2 + 5x_4^2 + 4x_6^2 \geq 7$
13. $9x_3^2 + 4x_5^2 + 4x_6^2 \geq 7$
14. $5x_1 + 4x_5 + 4x_6 \geq 7$
15. $5x_2 + 3x_3 + 5x_4 \geq 7$
16. $5x_1^2 + 5x_2^2 + 9x_3^2 \geq 7$
17. $5x_1 + 5x_2 + 4x_6 \geq 7$
18. $5x_1 + 5x_2 + 4x_5 \geq 7$
19. $9x_3^2 + 5x_4^2 + 4x_5^2 \geq 7$
20. $5x_1^2 + 5x_4^2 + 4x_6^2 \geq 7$
21. $5x_4 + 4x_5 + 4x_6 \geq 7$
22. $5x_2 + 5x_4 + 4x_6 \geq 7$
23. $9x_3^2 + 4x_5^2 + 4x_6^2 \geq 8$
24. $-9x_1 + 7x_6 \geq 0$
25. $-7x_3 + 10x_5 \geq 0$
26. $8x_1 - 9x_5 \geq 0$
27. $-7x_4 + 6x_5 \geq 0$
28. $5x_2 + 5x_4 + 4x_6 \leq 21$

## 4: Define variable types
- $x_1$ : Continuous
- $x_2$ : Integer
- $x_3$ : Integer
- $x_4$ : Integer
- $x_5$ : Integer
- $x_6$ : Continuous

## 5: Symbolic Representation
```json
{
    'sym_variables': [
        ('x1', 'cherry pies'),
        ('x2', 'knishes'),
        ('x3', 'chicken thighs'),
        ('x4', 'fruit salads'),
        ('x5', 'bowls of instant ramen'),
        ('x6', 'hot dogs')
    ],
    'objective_function': '4*x1^2 + 4*x1*x2 + 9*x2^2 + 8*x2*x4 + 3*x2*x5 + 8*x2*x6 + 9*x3^2 + x3*x4 + 6*x4*x5 + 2*x6^2 + 4*x1 + 5*x2 + 4*x3 + 8*x6',
    'constraints': [
        '5*x1 + 5*x2 + 3*x3 + 5*x4 + 4*x5 + 4*x6 <= 51',
        '5*x2 + 4*x6 >= 3',
        '5*x1 + 5*x2 >= 2',
        '5*x1 + 4*x5 + 4*x6 >= 7',
        '5*x2 + 3*x3 + 5*x4 >= 7',
        '5*x1 + 5*x2 + 3*x3 >= 7',
        '5*x1^2 + 5*x2^2 + 4*x6^2 >= 7',
        '5*x1 + 5*x2 + 4*x5 >= 7',
        '3*x3 + 5*x4 + 4*x5 >= 7',
        '5*x1 + 5*x4 + 4*x6 >= 7',
        '5*x4^2 + 4*x5^2 + 4*x6^2 >= 7',
        '5*x2^2 + 5*x4^2 + 4*x6^2 >= 7',
        '9*x3^2 + 4*x5^2 + 4*x6^2 >= 7',
        '5*x1 + 4*x5 + 4*x6 >= 7',
        '5*x2 + 3*x3 + 5*x4 >= 7',
        '5*x1^2 + 5*x2^2 + 9*x3^2 >= 7',
        '5*x1 + 5*x2 + 4*x6 >= 7',
        '5*x1 + 5*x2 + 4*x5 >= 7',
        '9*x3^2 + 5*x4^2 + 4*x5^2 >= 7',
        '5*x1^2 + 5*x4^2 + 4*x6^2 >= 7',
        '5*x4 + 4*x5 + 4*x6 >= 7',
        '5*x2 + 5*x4 + 4*x6 >= 7',
        '9*x3^2 + 4*x5^2 + 4*x6^2 >= 8',
        '-9*x1 + 7*x6 >= 0',
        '-7*x3 + 10*x5 >= 0',
        '8*x1 - 9*x5 >= 0',
        '-7*x4 + 6*x5 >= 0',
        '5*x2 + 5*x4 + 4*x6 <= 21'
    ]
}
```

## 6: Gurobi Code
```python
import gurobi

def optimization_problem():
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="cherry_pies", vtype=gurobi.GRB.CONTINUOUS)
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="knishes", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="chicken_thighs", vtype=gurobi.GRB.INTEGER)
    x4 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="fruit_salads", vtype=gurobi.GRB.INTEGER)
    x5 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="bowls_of_instant_ramen", vtype=gurobi.GRB.INTEGER)
    x6 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="hot_dogs", vtype=gurobi.GRB.CONTINUOUS)

    # Objective function
    model.setObjective(4*x1**2 + 4*x1*x2 + 9*x2**2 + 8*x2*x4 + 3*x2*x5 + 8*x2*x6 + 9*x3**2 + x3*x4 + 6*x4*x5 + 2*x6**2 + 4*x1 + 5*x2 + 4*x3 + 8*x6, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(5*x1 + 5*x2 + 3*x3 + 5*x4 + 4*x5 + 4*x6 <= 51)
    model.addConstr(5*x2 + 4*x6 >= 3)
    model.addConstr(5*x1 + 5*x2 >= 2)
    model.addConstr(5*x1 + 4*x5 + 4*x6 >= 7)
    model.addConstr(5*x2 + 3*x3 + 5*x4 >= 7)
    model.addConstr(5*x1 + 5*x2 + 3*x3 >= 7)
    model.addConstr(5*x1**2 + 5*x2**2 + 4*x6**2 >= 7)
    model.addConstr(5*x1 + 5*x2 + 4*x5 >= 7)
    model.addConstr(3*x3 + 5*x4 + 4*x5 >= 7)
    model.addConstr(5*x1 + 5*x4 + 4*x6 >= 7)
    model.addConstr(5*x4**2 + 4*x5**2 + 4*x6**2 >= 7)
    model.addConstr(5*x2**2 + 5*x4**2 + 4*x6**2 >= 7)
    model.addConstr(9*x3**2 + 4*x5**2 + 4*x6**2 >= 7)
    model.addConstr(5*x1 + 4*x5 + 4*x6 >= 7)
    model.addConstr(5*x2 + 3*x3 + 5*x4 >= 7)
    model.addConstr(5*x1**2 + 5*x2**2 + 9*x3**2 >= 7)
    model.addConstr(5*x1 + 5*x2 + 4*x6 >= 7)
    model.addConstr(5*x1 + 5*x2 + 4*x5 >= 7)
    model.addConstr(9*x3**2 + 5*x4**2 + 4*x5**2 >= 7)
    model.addConstr(5*x1**2 + 5*x4**2 + 4*x6**2 >= 7)
    model.addConstr(5*x4 + 4*x5 + 4*x6 >= 7)
    model.addConstr(5*x2 + 5*x4 + 4*x6 >= 7)
    model.addConstr(9*x3**2 + 4*x5**2 + 4*x6**2 >= 8)
    model.addConstr(-9*x1 + 7*x6 >= 0)
    model.addConstr(-7*x3 + 10*x5 >= 0)
    model.addConstr(8*x1 - 9*x5 >= 0)
    model.addConstr(-7*x4 + 6*x5 >= 0)
    model.addConstr(5*x2 + 5*x4 + 4*x6 <= 21)

    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print("Objective: ", model.objVal)
        print("x1: ", x1.varValue)
        print("x2: ", x2.varValue)
        print("x3: ", x3.varValue)
        print("x4: ", x4.varValue)
        print("x5: ", x5.varValue)
        print("x6: ", x6.varValue)
    else:
        print("No optimal solution found.")

optimization_problem()
```