## Step 1: Define the variables and their productivity ratings
Let's denote the hours worked by John, Peggy, Paul, Bobby, Mary, Laura, and Jean as $J$, $P$, $L$, $B$, $M$, $LA$, and $JE$ respectively. Their productivity ratings are given as: $r_J = 22$, $r_P = 17$, $r_L = 4$, $r_B = 29$, $r_M = 18$, $r_{LA} = 15$, and $r_{JE} = 22$.

## 2: Formulate the objective function
The objective function to maximize is: $3J^2 + 4P \cdot M + 3L \cdot M + 4L \cdot LA + 7B \cdot LA + M \cdot LA + 3B$.

## 3: Express the objective function in terms of Gurobi variables
We will define the variables in Gurobi and then formulate the objective function.

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

## 5: Implement the problem in Gurobi
```python
import gurobipy as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
J = m.addVar(name="hours_worked_by_John", lb=-gp.GRB.INFINITY)
P = m.addVar(name="hours_worked_by_Peggy", lb=-gp.GRB.INFINITY)
L = m.addVar(name="hours_worked_by_Paul", lb=-gp.GRB.INFINITY)
B = m.addVar(name="hours_worked_by_Bobby", lb=-gp.GRB.INFINITY)
M = m.addVar(name="hours_worked_by_Mary", lb=-gp.GRB.INFINITY)
LA = m.addVar(name="hours_worked_by_Laura", lb=-gp.GRB.INFINITY)
JE = m.addVar(name="hours_worked_by_Jean", lb=-gp.GRB.INFINITY)

# Objective function
m.setObjective(3*J**2 + 4*P*M + 3*L*M + 4*L*LA + 7*B*LA + M*LA + 3*B, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(-10*J**2 + 10*P**2 >= 0)
m.addConstr(B**2 + M**2 <= 385)
m.addConstr(P**2 + JE**2 <= 449)
m.addConstr(P + M <= 432)
m.addConstr(P + L <= 855)
m.addConstr(P**2 + B**2 <= 264)
m.addConstr(J**2 + M**2 <= 364)
m.addConstr(M + LA <= 283)
m.addConstr(L + JE <= 363)
m.addConstr(L + M <= 653)
m.addConstr(J + LA <= 253)
m.addConstr(J + L <= 690)
m.addConstr(J + P <= 300)
m.addConstr(J + JE <= 279)
m.addConstr(M**2 + JE**2 <= 527)
m.addConstr(J + B <= 199)
m.addConstr(J**2 + LA**2 + JE**2 <= 571)
m.addConstr(P + L + LA <= 527)
m.addConstr(J + P + JE <= 650)
m.addConstr(L + M + LA <= 862)
m.addConstr(B + LA + JE <= 881)
m.addConstr(J + P + LA <= 561)
m.addConstr(P + L + M <= 817)
m.addConstr(J + L + LA <= 402)
m.addConstr(P + B + JE <= 698)
m.addConstr(J + B + JE <= 380)
m.addConstr(L + B + M <= 824)
m.addConstr(B + M + LA <= 699)
m.addConstr(P + B + M <= 271)
m.addConstr(L + B + JE <= 899)
m.addConstr(L + B + LA <= 199)
m.addConstr(J + L + B <= 225)
m.addConstr(J + P + L + B + M + LA + JE <= 225)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.Status.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by John: ", J.varValue)
    print("Hours worked by Peggy: ", P.varValue)
    print("Hours worked by Paul: ", L.varValue)
    print("Hours worked by Bobby: ", B.varValue)
    print("Hours worked by Mary: ", M.varValue)
    print("Hours worked by Laura: ", LA.varValue)
    print("Hours worked by Jean: ", JE.varValue)
else:
    print("The model is infeasible")
```