## Step 1: Define the variables and their corresponding symbolic notation
Let's denote the variables as follows:
- $x_1$ = hours worked by Peggy
- $x_2$ = hours worked by Hank
- $x_3$ = hours worked by Bill
- $x_4$ = hours worked by John
- $x_5$ = hours worked by Jean

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is:
\[3x_1^2 + 5x_1x_3 + 8x_2x_3 + 7x_2x_4 + 6x_2x_5 + 5x_4\]

## 3: List the constraints in symbolic notation
Constraints:
1. $15x_1$ = 15 (Peggy's paperwork competence rating)
2. $4x_2$ = 4 (Hank's paperwork competence rating)
3. $16x_3$ = 16 (Bill's paperwork competence rating)
4. $14x_4$ = 14 (John's paperwork competence rating)
5. $5x_5$ = 5 (Jean's paperwork competence rating)
6. $4x_2 + 5x_5 \geq 23$
7. $4x_2 + 16x_3 \geq 14$
8. $4x_2 + 16x_3 + 5x_5 \geq 26$
9. $4x_2 + 14x_4 + 5x_5 \geq 26$
10. $15^2x_1^2 + 4^2x_2^2 + 5^2x_5^2 \geq 26$
11. $15^2x_1^2 + 16^2x_3^2 + 5^2x_5^2 \geq 26$
12. $16^2x_3^2 + 14^2x_4^2 + 5^2x_5^2 \geq 26$
13. $4x_2 + 16x_3 + 5x_5 \geq 17$
14. $4x_2 + 14x_4 + 5x_5 \geq 17$
15. $15x_1 + 4x_2 + 5x_5 \geq 17$
16. $15x_1 + 16x_3 + 5x_5 \geq 17$
17. $16x_3 + 14x_4 + 5x_5 \geq 17$
18. $4x_2 + 16x_3 + 5x_5 \geq 18$
19. $4^2x_2^2 + 14^2x_4^2 + 5^2x_5^2 \geq 18$
20. $15x_1 + 4x_2 + 5x_5 \geq 18$
21. $15^2x_1^2 + 16^2x_3^2 + 5^2x_5^2 \geq 18$
22. $16x_3 + 14x_4 + 5x_5 \geq 18$
23. $4x_2 + 16x_3 + 5x_5 \geq 15$
24. $4x_2 + 14x_4 + 5x_5 \geq 15$
25. $15^2x_1^2 + 4^2x_2^2 + 5^2x_5^2 \geq 15$
26. $15^2x_1^2 + 16^2x_3^2 + 5^2x_5^2 \geq 15$
27. $16x_3 + 14x_4 + 5x_5 \geq 15$
28. $4x_2 + 16x_3 + 5x_5 \geq 18$
29. $4x_2 + 14x_4 + 5x_5 \geq 18$
30. $15x_1 + 4x_2 + 5x_5 \geq 18$
31. $15x_1 + 16x_3 + 5x_5 \geq 18$
32. $16x_3 + 14x_4 + 5x_5 \geq 18$
33. $15x_1 + 4x_2 + 16x_3 + 14x_4 + 5x_5 \geq 18$
34. $x_2 - 9x_5 \geq 0$
35. $5x_1 - 4x_4 \geq 0$
36. $-6x_3^2 + 6x_5^2 \geq 0$
37. $4^2x_2^2 + 14^2x_4^2 \leq 74$
38. $16x_3 + 5x_5 \leq 59$
39. $15x_1 + 5x_5 \leq 42$
40. $15x_1 + 4x_2 \leq 52$
41. $14x_4 + 5x_5 \leq 58$
42. $15x_1 + 16x_3 \leq 129$
43. $15x_1 + 14x_4 \leq 145$
44. $16^2x_3^2 + 14^2x_4^2 \leq 82$
45. $4x_2 + 5x_5 \leq 35$
46. $4x_2 + 14x_4 + 5x_5 \leq 84$
47. $15x_1 + 4x_2 + 14x_4 \leq 75$
48. $x_1$ is an integer
49. $x_2$ is an integer
50. $x_5$ is an integer

## 4: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Peggy'),
        ('x2', 'hours worked by Hank'),
        ('x3', 'hours worked by Bill'),
        ('x4', 'hours worked by John'),
        ('x5', 'hours worked by Jean')
    ],
    'objective_function': '3*x1^2 + 5*x1*x3 + 8*x2*x3 + 7*x2*x4 + 6*x2*x5 + 5*x4',
    'constraints': [
        '15*x1 = 15',
        '4*x2 = 4',
        '16*x3 = 16',
        '14*x4 = 14',
        '5*x5 = 5',
        '4*x2 + 5*x5 >= 23',
        '4*x2 + 16*x3 >= 14',
        '4*x2 + 16*x3 + 5*x5 >= 26',
        '4*x2 + 14*x4 + 5*x5 >= 26',
        '15^2*x1^2 + 4^2*x2^2 + 5^2*x5^2 >= 26',
        '15^2*x1^2 + 16^2*x3^2 + 5^2*x5^2 >= 26',
        '16^2*x3^2 + 14^2*x4^2 + 5^2*x5^2 >= 26',
        '4*x2 + 16*x3 + 5*x5 >= 17',
        '4*x2 + 14*x4 + 5*x5 >= 17',
        '15*x1 + 4*x2 + 5*x5 >= 17',
        '15*x1 + 16*x3 + 5*x5 >= 17',
        '16*x3 + 14*x4 + 5*x5 >= 17',
        '4*x2 + 16*x3 + 5*x5 >= 18',
        '4^2*x2^2 + 14^2*x4^2 + 5^2*x5^2 >= 18',
        '15*x1 + 4*x2 + 5*x5 >= 18',
        '15^2*x1^2 + 16^2*x3^2 + 5^2*x5^2 >= 18',
        '16*x3 + 14*x4 + 5*x5 >= 18',
        '4*x2 + 16*x3 + 5*x5 >= 15',
        '4*x2 + 14*x4 + 5*x5 >= 15',
        '15^2*x1^2 + 4^2*x2^2 + 5^2*x5^2 >= 15',
        '15^2*x1^2 + 16^2*x3^2 + 5^2*x5^2 >= 15',
        '16*x3 + 14*x4 + 5*x5 >= 15',
        '4*x2 + 16*x3 + 5*x5 >= 18',
        '4*x2 + 14*x4 + 5*x5 >= 18',
        '15*x1 + 4*x2 + 5*x5 >= 18',
        '15*x1 + 16*x3 + 5*x5 >= 18',
        '16*x3 + 14*x4 + 5*x5 >= 18',
        '15*x1 + 4*x2 + 16*x3 + 14*x4 + 5*x5 >= 18',
        'x2 - 9*x5 >= 0',
        '5*x1 - 4*x4 >= 0',
        '-6*x3^2 + 6*x5^2 >= 0',
        '4^2*x2^2 + 14^2*x4^2 <= 74',
        '16*x3 + 5*x5 <= 59',
        '15*x1 + 5*x5 <= 42',
        '15*x1 + 4*x2 <= 52',
        '14*x4 + 5*x5 <= 58',
        '15*x1 + 16*x3 <= 129',
        '15*x1 + 14*x4 <= 145',
        '16^2*x3^2 + 14^2*x4^2 <= 82',
        '4*x2 + 5*x5 <= 35',
        '4*x2 + 14*x4 + 5*x5 <= 84',
        '15*x1 + 4*x2 + 14*x4 <= 75',
        'x1 % 1 == 0',
        'x2 % 1 == 0',
        'x5 % 1 == 0'
    ]
}
```

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

# Define the model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by Peggy
x2 = m.addVar(name="x2", vtype=gurobi.GRB.INTEGER)  # hours worked by Hank
x3 = m.addVar(name="x3")  # hours worked by Bill
x4 = m.addVar(name="x4")  # hours worked by John
x5 = m.addVar(name="x5", vtype=gurobi.GRB.INTEGER)  # hours worked by Jean

# Objective function
m.setObjective(3*x1**2 + 5*x1*x3 + 8*x2*x3 + 7*x2*x4 + 6*x2*x5 + 5*x4, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(15*x1 == 15)
m.addConstr(4*x2 == 4)
m.addConstr(16*x3 == 16)
m.addConstr(14*x4 == 14)
m.addConstr(5*x5 == 5)
m.addConstr(4*x2 + 5*x5 >= 23)
m.addConstr(4*x2 + 16*x3 >= 14)
m.addConstr(4*x2 + 16*x3 + 5*x5 >= 26)
m.addConstr(4*x2 + 14*x4 + 5*x5 >= 26)
m.addConstr(15**2*x1**2 + 4**2*x2**2 + 5**2*x5**2 >= 26)
m.addConstr(15**2*x1**2 + 16**2*x3**2 + 5**2*x5**2 >= 26)
m.addConstr(16**2*x3**2 + 14**2*x4**2 + 5**2*x5**2 >= 26)
m.addConstr(4*x2 + 16*x3 + 5*x5 >= 17)
m.addConstr(4*x2 + 14*x4 + 5*x5 >= 17)
m.addConstr(15*x1 + 4*x2 + 5*x5 >= 17)
m.addConstr(15*x1 + 16*x3 + 5*x5 >= 17)
m.addConstr(16*x3 + 14*x4 + 5*x5 >= 17)
m.addConstr(4*x2 + 16*x3 + 5*x5 >= 18)
m.addConstr(4**2*x2**2 + 14**2*x4**2 + 5**2*x5**2 >= 18)
m.addConstr(15*x1 + 4*x2 + 5*x5 >= 18)
m.addConstr(15**2*x1**2 + 16**2*x3**2 + 5**2*x5**2 >= 18)
m.addConstr(16*x3 + 14*x4 + 5*x5 >= 18)
m.addConstr(4*x2 + 16*x3 + 5*x5 >= 15)
m.addConstr(4*x2 + 14*x4 + 5*x5 >= 15)
m.addConstr(15**2*x1**2 + 4**2*x2**2 + 5**2*x5**2 >= 15)
m.addConstr(15**2*x1**2 + 16**2*x3**2 + 5**2*x5**2 >= 15)
m.addConstr(16*x3 + 14*x4 + 5*x5 >= 15)
m.addConstr(4*x2 + 16*x3 + 5*x5 >= 18)
m.addConstr(4*x2 + 14*x4 + 5*x5 >= 18)
m.addConstr(15*x1 + 4*x2 + 5*x5 >= 18)
m.addConstr(15*x1 + 16*x3 + 5*x5 >= 18)
m.addConstr(16*x3 + 14*x4 + 5*x5 >= 18)
m.addConstr(15*x1 + 4*x2 + 16*x3 + 14*x4 + 5*x5 >= 18)
m.addConstr(x2 - 9*x5 >= 0)
m.addConstr(5*x1 - 4*x4 >= 0)
m.addConstr(-6*x3**2 + 6*x5**2 >= 0)
m.addConstr(4**2*x2**2 + 14**2*x4**2 <= 74)
m.addConstr(16*x3 + 5*x5 <= 59)
m.addConstr(15*x1 + 5*x5 <= 42)
m.addConstr(15*x1 + 4*x2 <= 52)
m.addConstr(14*x4 + 5*x5 <= 58)
m.addConstr(15*x1 + 16*x3 <= 129)
m.addConstr(15*x1 + 14*x4 <= 145)
m.addConstr(16**2*x3**2 + 14**2*x4**2 <= 82)
m.addConstr(4*x2 + 5*x5 <= 35)
m.addConstr(4*x2 + 14*x4 + 5*x5 <= 84)
m.addConstr(15*x1 + 4*x2 + 14*x4 <= 75)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
    print("x4: ", x4.varValue)
    print("x5: ", x5.varValue)
else:
    print("The model is infeasible")
```