## Step 1: Define the optimization problem
The goal is to maximize the objective function: $5.02 \times (hours\_worked\_by\_Bill)^2 + 4.37 \times hours\_worked\_by\_Bill \times hours\_worked\_by\_John + 3.47 \times (hours\_worked\_by\_John)^2 + 3.78 \times hours\_worked\_by\_John$.

## Step 2: List all the constraints
1. Bill's dollar cost per hour is 8.
2. Bill's work quality rating is 2.
3. Bill's likelihood to quit index is 6.
4. John's dollar cost per hour is 13.
5. John's work quality rating is 9.
6. John's likelihood to quit index is 12.
7. $(hours\_worked\_by\_Bill)^2 + (hours\_worked\_by\_John)^2 \geq 35$.
8. $2 \times hours\_worked\_by\_Bill + 9 \times hours\_worked\_by\_John \geq 41$.
9. $6 \times hours\_worked\_by\_Bill + 12 \times hours\_worked\_by\_John \geq 38$.
10. $3 \times hours\_worked\_by\_Bill - 6 \times hours\_worked\_by\_John \geq 0$.
11. $(hours\_worked\_by\_Bill)^2 + (hours\_worked\_by\_John)^2 \leq 123$.
12. $8 \times hours\_worked\_by\_Bill + 13 \times hours\_worked\_by\_John \leq 123$.
13. $(hours\_worked\_by\_Bill)^2 + (hours\_worked\_by\_John)^2 \leq 80$ is incorrect and should be $2 \times (hours\_worked\_by\_Bill)^2 + 9 \times (hours\_worked\_by\_John)^2 \leq 80$ is also not correct, use $2*hours\_worked\_by\_Bill + 9*hours\_worked\_by\_John \leq 80$ for work quality rating.
14. $2 \times hours\_worked\_by\_Bill + 9 \times hours\_worked\_by\_John \leq 80$.
15. $6 \times hours\_worked\_by\_Bill + 12 \times hours\_worked\_by\_John \leq 71$.
16. $hours\_worked\_by\_Bill$ must be an integer.

## 3: Convert the problem into Gurobi code
```python
import gurobi

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

# Define the variables
hours_worked_by_Bill = m.addVar(name="hours_worked_by_Bill", integer=True)
hours_worked_by_John = m.addVar(name="hours_worked_by_John")

# Objective function
m.setObjective(5.02 * hours_worked_by_Bill ** 2 + 4.37 * hours_worked_by_Bill * hours_worked_by_John + 
               3.47 * hours_worked_by_John ** 2 + 3.78 * hours_worked_by_John, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(hours_worked_by_Bill ** 2 + hours_worked_by_John ** 2 >= 35)
m.addConstr(2 * hours_worked_by_Bill + 9 * hours_worked_by_John >= 41)
m.addConstr(6 * hours_worked_by_Bill + 12 * hours_worked_by_John >= 38)
m.addConstr(3 * hours_worked_by_Bill - 6 * hours_worked_by_John >= 0)
m.addConstr(hours_worked_by_Bill ** 2 + hours_worked_by_John ** 2 <= 123)
m.addConstr(8 * hours_worked_by_Bill + 13 * hours_worked_by_John <= 123)
m.addConstr(2 * hours_worked_by_Bill + 9 * hours_worked_by_John <= 80)
m.addConstr(6 * hours_worked_by_Bill + 12 * hours_worked_by_John <= 71)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Bill: {hours_worked_by_Bill.varValue}")
    print(f"Hours worked by John: {hours_worked_by_John.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```