## Step 1: Define the variables and their corresponding symbolic notation
The variables are: ['hours worked by John', 'hours worked by Peggy', 'hours worked by Paul', 'hours worked by Bobby', 'hours worked by Mary', 'hours worked by Laura', 'hours worked by Jean']. 
Let's denote them as: 
- $x_1$ for 'hours worked by John'
- $x_2$ for 'hours worked by Peggy'
- $x_3$ for 'hours worked by Paul'
- $x_4$ for 'hours worked by Bobby'
- $x_5$ for 'hours worked by Mary'
- $x_6$ for 'hours worked by Laura'
- $x_7$ for 'hours worked by Jean'

## Step 2: Define the productivity ratings
The productivity ratings are given as: 
- $r_0$ with $x_0 = 22, x_1 = 17, x_2 = 4, x_3 = 29, x_4 = 18, x_5 = 15, x_6 = 22$ for John, Peggy, Paul, Bobby, Mary, Laura, Jean respectively.

## 3: Formulate the objective function
The objective function to maximize is: $3x_1^2 + 4x_2x_5 + 3x_3x_5 + 4x_3x_6 + 7x_4x_6 + x_5x_6 + 3x_4$.

## 4: List all constraints
1. $-10x_1^2 + 10x_2^2 \geq 0$
2. $x_4^2 + x_5^2 \leq 385$
3. $x_2^2 + x_7^2 \leq 449$
4. $x_2 + x_5 \leq 432$
5. $x_2 + x_3 \leq 855$
6. $x_2^2 + x_4^2 \leq 264$
7. $x_1^2 + x_5^2 \leq 364$
8. $x_5 + x_6 \leq 283$
9. $x_3 + x_7 \leq 363$
10. $x_3 + x_5 \leq 653$
11. $x_1 + x_6 \leq 253$
12. $x_1 + x_3 \leq 690$
13. $x_1 + x_2 \leq 300$
14. $x_1 + x_7 \leq 279$
15. $x_5^2 + x_7^2 \leq 527$
16. $x_1 + x_4 \leq 199$
17. $x_1^2 + x_6^2 + x_7^2 \leq 571$
18. $x_2 + x_3 + x_6 \leq 527$
19. $x_1 + x_2 + x_7 \leq 650$
20. $x_3 + x_5 + x_6 \leq 862$
21. $x_4 + x_6 + x_7 \leq 881$
22. $x_1 + x_2 + x_6 \leq 561$
23. $x_2 + x_3 + x_5 \leq 817$
24. $x_1 + x_3 + x_6 \leq 402$
25. $x_2 + x_4 + x_7 \leq 698$
26. $x_1 + x_4 + x_7 \leq 380$
27. $x_3 + x_4 + x_5 \leq 824$
28. $x_4 + x_5 + x_6 \leq 699$
29. $x_2 + x_4 + x_5 \leq 271$
30. $x_3 + x_4 + x_7 \leq 899$
31. $x_3 + x_4 + x_6 \leq 199$
32. $x_1 + x_3 + x_4 \leq 225$
33. $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 \leq 225$

## 5: Symbolic Representation
```json
{
    'sym_variables': [
        ('x1', 'hours worked by John'), 
        ('x2', 'hours worked by Peggy'), 
        ('x3', 'hours worked by Paul'), 
        ('x4', 'hours worked by Bobby'), 
        ('x5', 'hours worked by Mary'), 
        ('x6', 'hours worked by Laura'), 
        ('x7', 'hours worked by Jean')
    ], 
    'objective_function': '3*x1^2 + 4*x2*x5 + 3*x3*x5 + 4*x3*x6 + 7*x4*x6 + x5*x6 + 3*x4', 
    'constraints': [
        '-10*x1^2 + 10*x2^2 >= 0',
        'x4^2 + x5^2 <= 385',
        'x2^2 + x7^2 <= 449',
        'x2 + x5 <= 432',
        'x2 + x3 <= 855',
        'x2^2 + x4^2 <= 264',
        'x1^2 + x5^2 <= 364',
        'x5 + x6 <= 283',
        'x3 + x7 <= 363',
        'x3 + x5 <= 653',
        'x1 + x6 <= 253',
        'x1 + x3 <= 690',
        'x1 + x2 <= 300',
        'x1 + x7 <= 279',
        'x5^2 + x7^2 <= 527',
        'x1 + x4 <= 199',
        'x1^2 + x6^2 + x7^2 <= 571',
        'x2 + x3 + x6 <= 527',
        'x1 + x2 + x7 <= 650',
        'x3 + x5 + x6 <= 862',
        'x4 + x6 + x7 <= 881',
        'x1 + x2 + x6 <= 561',
        'x2 + x3 + x5 <= 817',
        'x1 + x3 + x6 <= 402',
        'x2 + x4 + x7 <= 698',
        'x1 + x4 + x7 <= 380',
        'x3 + x4 + x5 <= 824',
        'x4 + x5 + x6 <= 699',
        'x2 + x4 + x5 <= 271',
        'x3 + x4 + x7 <= 899',
        'x3 + x4 + x6 <= 199',
        'x1 + x3 + x4 <= 225',
        'x1 + x2 + x3 + x4 + x5 + x6 + x7 <= 225'
    ]
}
```

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

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

# Define variables
x1 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_John')
x2 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Peggy')
x3 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Paul')
x4 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Bobby')
x5 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Mary')
x6 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Laura')
x7 = m.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name='hours_worked_by_Jean')

# Objective function
m.setObjective(3*x1**2 + 4*x2*x5 + 3*x3*x5 + 4*x3*x6 + 7*x4*x6 + x5*x6 + 3*x4, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(-10*x1**2 + 10*x2**2 >= 0)
m.addConstr(x4**2 + x5**2 <= 385)
m.addConstr(x2**2 + x7**2 <= 449)
m.addConstr(x2 + x5 <= 432)
m.addConstr(x2 + x3 <= 855)
m.addConstr(x2**2 + x4**2 <= 264)
m.addConstr(x1**2 + x5**2 <= 364)
m.addConstr(x5 + x6 <= 283)
m.addConstr(x3 + x7 <= 363)
m.addConstr(x3 + x5 <= 653)
m.addConstr(x1 + x6 <= 253)
m.addConstr(x1 + x3 <= 690)
m.addConstr(x1 + x2 <= 300)
m.addConstr(x1 + x7 <= 279)
m.addConstr(x5**2 + x7**2 <= 527)
m.addConstr(x1 + x4 <= 199)
m.addConstr(x1**2 + x6**2 + x7**2 <= 571)
m.addConstr(x2 + x3 + x6 <= 527)
m.addConstr(x1 + x2 + x7 <= 650)
m.addConstr(x3 + x5 + x6 <= 862)
m.addConstr(x4 + x6 + x7 <= 881)
m.addConstr(x1 + x2 + x6 <= 561)
m.addConstr(x2 + x3 + x5 <= 817)
m.addConstr(x1 + x3 + x6 <= 402)
m.addConstr(x2 + x4 + x7 <= 698)
m.addConstr(x1 + x4 + x7 <= 380)
m.addConstr(x3 + x4 + x5 <= 824)
m.addConstr(x4 + x5 + x6 <= 699)
m.addConstr(x2 + x4 + x5 <= 271)
m.addConstr(x3 + x4 + x7 <= 899)
m.addConstr(x3 + x4 + x6 <= 199)
m.addConstr(x1 + x3 + x4 <= 225)
m.addConstr(x1 + x2 + x3 + x4 + x5 + x6 + x7 <= 225)

# Optimize
m.optimize()

# Print solution
if m.status == gurobi.GRB.Status.OPTIMAL:
    print('Objective: ', m.objVal)
    print('Hours worked by John: ', x1.varValue)
    print('Hours worked by Peggy: ', x2.varValue)
    print('Hours worked by Paul: ', x3.varValue)
    print('Hours worked by Bobby: ', x4.varValue)
    print('Hours worked by Mary: ', x5.varValue)
    print('Hours worked by Laura: ', x6.varValue)
    print('Hours worked by Jean: ', x7.varValue)
else:
    print('No solution found')
```