## Step 1: Define the variables and their corresponding natural language objects
Let's denote the hours worked by Ringo, George, Mary, and Peggy as $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is: $7x_1^2 + 6x_1x_2 + x_1x_3 + 8x_1x_4 + 3x_2^2 + 3x_2x_3 + 4x_2x_4 + 3x_3^2 + 6x_3x_4 + 8x_4^2 + 2x_1 + 3x_2 + 5x_3 + 4x_4$.

## Step 3: List the constraints in symbolic notation
1. $11x_1 \leq 303$
2. $x_1 \leq 174$
3. $14x_1 \leq 218$
4. $10x_2 \leq 303$
5. $2x_2 \leq 174$
6. $17x_2 \leq 218$
7. $17x_3 \leq 303$
8. $17x_3 \leq 174$
9. $16x_3 \leq 218$
10. $19x_4 \leq 303$
11. $18x_4 \leq 174$
12. $13x_4 \leq 218$
13. $11x_1 + 10x_2 \geq 39$
14. $11^2x_1^2 + 19^2x_4^2 \geq 61$
15. $10x_2 + 17x_3 \geq 68$
16. $11x_1 + 10x_2 + 17x_3 + 19x_4 \geq 68$
17. $x_1 + 2x_2 \geq 22$
18. $x_1 + 17x_3 \geq 14$
19. $17x_3 + 18x_4 \geq 15$
20. $x_1^2 + x_2^2 + 17^2x_3^2 \geq 43$
21. $x_1 + 17x_3 + 18x_4 \geq 43$
22. $x_1^2 + x_2^2 + 17^2x_3^2 \geq 25$
23. $x_1 + 17x_3 + 18x_4 \geq 25$
24. $x_1 + x_2 + x_3 + x_4 \geq 25$
25. $17x_2 + 13x_4 \geq 24$
26. $14^2x_1^2 + 17^2x_2^2 \geq 19$
27. $14x_1 + 17x_2 + 16x_3 \geq 40$
28. $14x_1 + 16x_3 + 13x_4 \geq 40$
29. $14^2x_1^2 + 17^2x_2^2 + 16^2x_3^2 \geq 27$
30. $14^2x_1^2 + 16^2x_3^2 + 13^2x_4^2 \geq 27$
31. $14x_1 + 17x_2 + 16x_3 + 13x_4 \geq 27$
32. $6x_1 - 5x_4 \geq 0$
33. $11x_1 + 17x_3 + 19x_4 \leq 299$
34. $10x_2 + 17x_3 + 19x_4 \leq 247$
35. $x_1 + 2x_2 \leq 160$
36. $17x_3 + 18x_4 \leq 115$
37. $14x_1 + 13x_4 \leq 190$
38. $17^2x_2^2 + 16^2x_3^2 \leq 164$
39. $x_3 \geq 0$ and $x_3$ is an integer.

## 4: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Ringo'),
        ('x2', 'hours worked by George'),
        ('x3', 'hours worked by Mary'),
        ('x4', 'hours worked by Peggy')
    ],
    'objective_function': '7*x1^2 + 6*x1*x2 + x1*x3 + 8*x1*x4 + 3*x2^2 + 3*x2*x3 + 4*x2*x4 + 3*x3^2 + 6*x3*x4 + 8*x4^2 + 2*x1 + 3*x2 + 5*x3 + 4*x4',
    'constraints': [
        '11*x1 <= 303',
        'x1 <= 174',
        '14*x1 <= 218',
        '10*x2 <= 303',
        '2*x2 <= 174',
        '17*x2 <= 218',
        '17*x3 <= 303',
        '17*x3 <= 174',
        '16*x3 <= 218',
        '19*x4 <= 303',
        '18*x4 <= 174',
        '13*x4 <= 218',
        '11*x1 + 10*x2 >= 39',
        '11^2*x1^2 + 19^2*x4^2 >= 61',
        '10*x2 + 17*x3 >= 68',
        '11*x1 + 10*x2 + 17*x3 + 19*x4 >= 68',
        'x1 + 2*x2 >= 22',
        'x1 + 17*x3 >= 14',
        '17*x3 + 18*x4 >= 15',
        'x1^2 + x2^2 + 17^2*x3^2 >= 43',
        'x1 + 17*x3 + 18*x4 >= 43',
        'x1^2 + x2^2 + 17^2*x3^2 >= 25',
        'x1 + 17*x3 + 18*x4 >= 25',
        'x1 + x2 + x3 + x4 >= 25',
        '17*x2 + 13*x4 >= 24',
        '14^2*x1^2 + 17^2*x2^2 >= 19',
        '14*x1 + 17*x2 + 16*x3 >= 40',
        '14*x1 + 16*x3 + 13*x4 >= 40',
        '14^2*x1^2 + 17^2*x2^2 + 16^2*x3^2 >= 27',
        '14^2*x1^2 + 16^2*x3^2 + 13^2*x4^2 >= 27',
        '14*x1 + 17*x2 + 16*x3 + 13*x4 >= 27',
        '6*x1 - 5*x4 >= 0',
        '11*x1 + 17*x3 + 19*x4 <= 299',
        '10*x2 + 17*x3 + 19*x4 <= 247',
        'x1 + 2*x2 <= 160',
        '17*x3 + 18*x4 <= 115',
        '14*x1 + 13*x4 <= 190',
        '17^2*x2^2 + 16^2*x3^2 <= 164',
        'x3 >= 0 and x3 % 1 == 0'
    ]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x1 = m.addVar(name="x1", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x2 = m.addVar(name="x2", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x3 = m.addVar(name="x3", lb=0, ub=gp.GRB.INFINITY, vtype=gp.GRB.INTEGER)
x4 = m.addVar(name="x4", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)

# Define the objective function
m.setObjective(7*x1**2 + 6*x1*x2 + x1*x3 + 8*x1*x4 + 3*x2**2 + 3*x2*x3 + 4*x2*x4 + 3*x3**2 + 6*x3*x4 + 8*x4**2 + 2*x1 + 3*x2 + 5*x3 + 4*x4, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(11*x1 <= 303)
m.addConstr(x1 <= 174)
m.addConstr(14*x1 <= 218)
m.addConstr(10*x2 <= 303)
m.addConstr(2*x2 <= 174)
m.addConstr(17*x2 <= 218)
m.addConstr(17*x3 <= 303)
m.addConstr(17*x3 <= 174)
m.addConstr(16*x3 <= 218)
m.addConstr(19*x4 <= 303)
m.addConstr(18*x4 <= 174)
m.addConstr(13*x4 <= 218)
m.addConstr(11*x1 + 10*x2 >= 39)
m.addConstr(11**2*x1**2 + 19**2*x4**2 >= 61)
m.addConstr(10*x2 + 17*x3 >= 68)
m.addConstr(11*x1 + 10*x2 + 17*x3 + 19*x4 >= 68)
m.addConstr(x1 + 2*x2 >= 22)
m.addConstr(x1 + 17*x3 >= 14)
m.addConstr(17*x3 + 18*x4 >= 15)
m.addConstr(x1**2 + x2**2 + 17**2*x3**2 >= 43)
m.addConstr(x1 + 17*x3 + 18*x4 >= 43)
m.addConstr(x1**2 + x2**2 + 17**2*x3**2 >= 25)
m.addConstr(x1 + 17*x3 + 18*x4 >= 25)
m.addConstr(x1 + x2 + x3 + x4 >= 25)
m.addConstr(17*x2 + 13*x4 >= 24)
m.addConstr(14**2*x1**2 + 17**2*x2**2 >= 19)
m.addConstr(14*x1 + 17*x2 + 16*x3 >= 40)
m.addConstr(14*x1 + 16*x3 + 13*x4 >= 40)
m.addConstr(14**2*x1**2 + 17**2*x2**2 + 16**2*x3**2 >= 27)
m.addConstr(14**2*x1**2 + 16**2*x3**2 + 13**2*x4**2 >= 27)
m.addConstr(14*x1 + 17*x2 + 16*x3 + 13*x4 >= 27)
m.addConstr(6*x1 - 5*x4 >= 0)
m.addConstr(11*x1 + 17*x3 + 19*x4 <= 299)
m.addConstr(10*x2 + 17*x3 + 19*x4 <= 247)
m.addConstr(x1 + 2*x2 <= 160)
m.addConstr(17*x3 + 18*x4 <= 115)
m.addConstr(14*x1 + 13*x4 <= 190)
m.addConstr(17**2*x2**2 + 16**2*x3**2 <= 164)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
    print("x4: ", x4.varValue)
else:
    print("The model is infeasible or unbounded.")
```