## Step 1: Define the variables and their corresponding symbolic notation
Let's denote the variables as follows:
- 'hours worked by Laura' as $x_1$
- 'hours worked by Ringo' as $x_2$
- 'hours worked by John' as $x_3$
- 'hours worked by Paul' as $x_4$
- 'hours worked by Hank' as $x_5$
- 'hours worked by George' as $x_6$

## Step 2: Express the objective function using symbolic variables
The objective function to maximize is:
$9.65x_1x_2 + 7.47x_1x_3 + 6.82x_1x_5 + 6.51x_1x_6 + 7.67x_2^2 + 3.64x_2x_5 + 1.08x_2x_6 + 9.24x_3^2 + 9.39x_3x_5 + 1.46x_3x_6 + 8.24x_4^2 + 7.27x_4x_6 + 9.2x_5^2 + 6.82x_5x_6 + 1.5x_6^2 + 9.53x_2 + 1.88x_3 + 8.03x_5 + 6.24x_6$

## 3: List the constraints
Constraints:
1. $x_1 = 6$ (Laura's organization score)
2. $x_2 = 9$ (Ringo's organization score)
3. $x_3 = 11$ (John's organization score)
4. $x_4 = 1$ (Paul's organization score)
5. $x_5 = 7$ (Hank's organization score)
6. $x_6 = 10$ (George's organization score)
7. $x_2^2 + x_4^2 + x_6^2 \geq 15$
8. $x_1 + x_3 + x_5 \geq 15$
9. $x_1^2 + x_2^2 + x_4^2 \geq 15$
10. $x_2^2 + x_4^2 + x_6^2 \geq 18$
11. $x_1 + x_3 + x_5 \geq 18$
12. $x_1^2 + x_2^2 + x_4^2 \geq 18$
13. $x_2^2 + x_4^2 + x_6^2 \geq 10$
14. $x_1 + x_3 + x_5 \geq 10$
15. $x_1^2 + x_2^2 + x_4^2 \geq 10$
16. $6x_3 - 3x_6 \geq 0$
17. $x_2^2 + x_4^2 \leq 47$
18. $x_2^2 + x_6^2 \leq 63$
19. $x_1 + x_6 \leq 56$
20. $x_5 + x_6 \leq 111$
21. $x_1^2 + x_2^2 \leq 62$
22. $x_1 + x_3 \leq 110$
23. $x_3 + x_4 \leq 28$
24. $x_2 + x_3 + x_4 \leq 97$
25. $x_1 + x_4 + x_5 \leq 74$
26. $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 \leq 74$

## 4: Symbolic Representation
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Laura'),
        ('x2', 'hours worked by Ringo'),
        ('x3', 'hours worked by John'),
        ('x4', 'hours worked by Paul'),
        ('x5', 'hours worked by Hank'),
        ('x6', 'hours worked by George')
    ],
    'objective_function': '9.65*x1*x2 + 7.47*x1*x3 + 6.82*x1*x5 + 6.51*x1*x6 + 7.67*x2^2 + 3.64*x2*x5 + 1.08*x2*x6 + 9.24*x3^2 + 9.39*x3*x5 + 1.46*x3*x6 + 8.24*x4^2 + 7.27*x4*x6 + 9.2*x5^2 + 6.82*x5*x6 + 1.5*x6^2 + 9.53*x2 + 1.88*x3 + 8.03*x5 + 6.24*x6',
    'constraints': [
        'x1 = 6',
        'x2 = 9',
        'x3 = 11',
        'x4 = 1',
        'x5 = 7',
        'x6 = 10',
        'x2^2 + x4^2 + x6^2 >= 15',
        'x1 + x3 + x5 >= 15',
        'x1^2 + x2^2 + x4^2 >= 15',
        'x2^2 + x4^2 + x6^2 >= 18',
        'x1 + x3 + x5 >= 18',
        'x1^2 + x2^2 + x4^2 >= 18',
        'x2^2 + x4^2 + x6^2 >= 10',
        'x1 + x3 + x5 >= 10',
        'x1^2 + x2^2 + x4^2 >= 10',
        '6*x3 - 3*x6 >= 0',
        'x2^2 + x4^2 <= 47',
        'x2^2 + x6^2 <= 63',
        'x1 + x6 <= 56',
        'x5 + x6 <= 111',
        'x1^2 + x2^2 <= 62',
        'x1 + x3 <= 110',
        'x3 + x4 <= 28',
        'x2 + x3 + x4 <= 97',
        'x1 + x4 + x5 <= 74',
        'x1 + x2 + x3 + x4 + x5 + x6 <= 74'
    ]
}
```

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

def optimization_problem():
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(lb=0, ub=None, vtype=gurobi.GRB.INTEGER, name="x1")  # Laura
    x2 = model.addVar(lb=0, ub=None, vtype=gurobi.GRB.INTEGER, name="x2")  # Ringo
    x3 = model.addVar(lb=0, ub=None, vtype=gurobi.GRB.CONTINUOUS, name="x3")  # John
    x4 = model.addVar(lb=0, ub=None, vtype=gurobi.GRB.INTEGER, name="x4")  # Paul
    x5 = model.addVar(lb=0, ub=None, vtype=gurobi.GRB.INTEGER, name="x5")  # Hank
    x6 = model.addVar(lb=0, ub=None, vtype=gurobi.GRB.INTEGER, name="x6")  # George

    # Set initial values
    x1.Start = 6
    x2.Start = 9
    x3.Start = 11
    x4.Start = 1
    x5.Start = 7
    x6.Start = 10

    # Objective function
    model.setObjective(9.65*x1*x2 + 7.47*x1*x3 + 6.82*x1*x5 + 6.51*x1*x6 + 7.67*x2**2 + 3.64*x2*x5 + 1.08*x2*x6 + 9.24*x3**2 + 9.39*x3*x5 + 1.46*x3*x6 + 8.24*x4**2 + 7.27*x4*x6 + 9.2*x5**2 + 6.82*x5*x6 + 1.5*x6**2 + 9.53*x2 + 1.88*x3 + 8.03*x5 + 6.24*x6, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x1 == 6)
    model.addConstr(x2 == 9)
    model.addConstr(x3 == 11)
    model.addConstr(x4 == 1)
    model.addConstr(x5 == 7)
    model.addConstr(x6 == 10)
    model.addConstr(x2**2 + x4**2 + x6**2 >= 15)
    model.addConstr(x1 + x3 + x5 >= 15)
    model.addConstr(x1**2 + x2**2 + x4**2 >= 15)
    model.addConstr(x2**2 + x4**2 + x6**2 >= 18)
    model.addConstr(x1 + x3 + x5 >= 18)
    model.addConstr(x1**2 + x2**2 + x4**2 >= 18)
    model.addConstr(x2**2 + x4**2 + x6**2 >= 10)
    model.addConstr(x1 + x3 + x5 >= 10)
    model.addConstr(x1**2 + x2**2 + x4**2 >= 10)
    model.addConstr(6*x3 - 3*x6 >= 0)
    model.addConstr(x2**2 + x4**2 <= 47)
    model.addConstr(x2**2 + x6**2 <= 63)
    model.addConstr(x1 + x6 <= 56)
    model.addConstr(x5 + x6 <= 111)
    model.addConstr(x1**2 + x2**2 <= 62)
    model.addConstr(x1 + x3 <= 110)
    model.addConstr(x3 + x4 <= 28)
    model.addConstr(x2 + x3 + x4 <= 97)
    model.addConstr(x1 + x4 + x5 <= 74)
    model.addConstr(x1 + x2 + x3 + x4 + x5 + x6 <= 74)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("x1: ", x1.varValue)
        print("x2: ", x2.varValue)
        print("x3: ", x3.varValue)
        print("x4: ", x4.varValue)
        print("x5: ", x5.varValue)
        print("x6: ", x6.varValue)
    else:
        print("No solution found")

optimization_problem()
```