## Step 1: Define the variables and the objective function
The variables are ['hours worked by Bobby', 'hours worked by Bill', 'hours worked by Peggy', 'hours worked by Mary', 'hours worked by Paul', 'hours worked by John']. Let's denote them as $x_1, x_2, x_3, x_4, x_5, x_6$ respectively.

## Step 2: Translate the objective function into algebraic terms
The objective function to maximize is $6x_1x_2 + 3x_1x_3 + 2x_1x_5 + 9x_3x_6 + 9x_4x_6 + 8x_5x_6 + 3x_1 + 3x_2 + 2x_5$.

## 3: List the constraints
Constraints include:
- $19x_1 \leq 657$ (likelihood to quit index for Bobby)
- $12x_1 \leq 837$ (computer competence rating for Bobby)
- $4x_1 \leq 268$ (work quality rating for Bobby)
- $22x_2 \leq 657$ (likelihood to quit index for Bill)
- $5x_2 \leq 837$ (computer competence rating for Bill)
- $20x_2 \leq 268$ (work quality rating for Bill)
- $22x_3 \leq 657$ (likelihood to quit index for Peggy)
- $20x_3 \leq 837$ (computer competence rating for Peggy)
- $5x_3 \leq 268$ (work quality rating for Peggy)
- $18x_4 \leq 657$ (likelihood to quit index for Mary)
- $11x_4 \leq 837$ (computer competence rating for Mary)
- $12x_4 \leq 268$ (work quality rating for Mary)
- $7x_5 \leq 657$ (likelihood to quit index for Paul)
- $6x_5 \leq 837$ (computer competence rating for Paul)
- $4x_5 \leq 268$ (work quality rating for Paul)
- $3x_6 \leq 657$ (likelihood to quit index for John)
- $21x_6 \leq 837$ (computer competence rating for John)
- $22x_6 \leq 268$ (work quality rating for John)
- $22x_3 + 7x_5 \geq 51$ (combined likelihood to quit index for Peggy and Paul)
- $x_2^2 + x_3^2 \geq 79$ (combined likelihood to quit index for Bill and Peggy)
- $18x_4 + 7x_5 \geq 78$ (combined likelihood to quit index for Mary and Paul)
- $19x_1 + 7x_5 + 3x_6 \geq 56$ (combined likelihood to quit index for Bobby, Paul, and John)
- $22x_3 + 7x_5 + 3x_6 \geq 56$ (combined likelihood to quit index for Peggy, Paul, and John)
- $18x_4 + 22x_3 + 3x_6 \geq 56$ (combined likelihood to quit index for Peggy, Mary, and John)
- $18x_4^2 + 7x_5^2 + 3x_6^2 \geq 56$ (combined likelihood to quit index for Mary, Paul, and John)
- $19x_1^2 + 22x_3^2 + 18x_4^2 \geq 56$ (combined likelihood to quit index for Bobby, Peggy, and Mary)
- $x_2^2 + x_3^2 + 18x_4^2 \geq 56$ (combined likelihood to quit index for Bill, Peggy, and Mary)
- $x_1, x_2, x_4, x_6 \in \mathbb{Z}$ (integer constraints)
- $x_3, x_5$ can be fractional

## 4: Symbolic Representation
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Bobby'), 
        ('x2', 'hours worked by Bill'), 
        ('x3', 'hours worked by Peggy'), 
        ('x4', 'hours worked by Mary'), 
        ('x5', 'hours worked by Paul'), 
        ('x6', 'hours worked by John')
    ], 
    'objective_function': '6*x1*x2 + 3*x1*x3 + 2*x1*x5 + 9*x3*x6 + 9*x4*x6 + 8*x5*x6 + 3*x1 + 3*x2 + 2*x5', 
    'constraints': [
        '19*x1 <= 657', '12*x1 <= 837', '4*x1 <= 268',
        '22*x2 <= 657', '5*x2 <= 837', '20*x2 <= 268',
        '22*x3 <= 657', '20*x3 <= 837', '5*x3 <= 268',
        '18*x4 <= 657', '11*x4 <= 837', '12*x4 <= 268',
        '7*x5 <= 657', '6*x5 <= 837', '4*x5 <= 268',
        '3*x6 <= 657', '21*x6 <= 837', '22*x6 <= 268',
        '22*x3 + 7*x5 >= 51', 'x2^2 + x3^2 >= 79',
        '18*x4 + 7*x5 >= 78', '19*x1 + 7*x5 + 3*x6 >= 56',
        '22*x3 + 7*x5 + 3*x6 >= 56', '18*x4 + 22*x3 + 3*x6 >= 56',
        '18*x4^2 + 7*x5^2 + 3*x6^2 >= 56', '19*x1^2 + 22*x3^2 + 18*x4^2 >= 56',
        'x2^2 + x3^2 + 18*x4^2 >= 56'
    ]
}
```

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

# Define the model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(name='x1', vtype=gurobi.GRB.INTEGER)  # hours worked by Bobby
x2 = m.addVar(name='x2', vtype=gurobi.GRB.INTEGER)  # hours worked by Bill
x3 = m.addVar(name='x3')  # hours worked by Peggy
x4 = m.addVar(name='x4', vtype=gurobi.GRB.INTEGER)  # hours worked by Mary
x5 = m.addVar(name='x5')  # hours worked by Paul
x6 = m.addVar(name='x6')  # hours worked by John

# Objective function
m.setObjective(6*x1*x2 + 3*x1*x3 + 2*x1*x5 + 9*x3*x6 + 9*x4*x6 + 8*x5*x6 + 3*x1 + 3*x2 + 2*x5, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(19*x1 <= 657)
m.addConstr(12*x1 <= 837)
m.addConstr(4*x1 <= 268)
m.addConstr(22*x2 <= 657)
m.addConstr(5*x2 <= 837)
m.addConstr(20*x2 <= 268)
m.addConstr(22*x3 <= 657)
m.addConstr(20*x3 <= 837)
m.addConstr(5*x3 <= 268)
m.addConstr(18*x4 <= 657)
m.addConstr(11*x4 <= 837)
m.addConstr(12*x4 <= 268)
m.addConstr(7*x5 <= 657)
m.addConstr(6*x5 <= 837)
m.addConstr(4*x5 <= 268)
m.addConstr(3*x6 <= 657)
m.addConstr(21*x6 <= 837)
m.addConstr(22*x6 <= 268)
m.addConstr(22*x3 + 7*x5 >= 51)
m.addConstr(x2**2 + x3**2 >= 79)
m.addConstr(18*x4 + 7*x5 >= 78)
m.addConstr(19*x1 + 7*x5 + 3*x6 >= 56)
m.addConstr(22*x3 + 7*x5 + 3*x6 >= 56)
m.addConstr(18*x4 + 22*x3 + 3*x6 >= 56)
m.addConstr(18*x4**2 + 7*x5**2 + 3*x6**2 >= 56)
m.addConstr(19*x1**2 + 22*x3**2 + 18*x4**2 >= 56)
m.addConstr(x2**2 + x3**2 + 18*x4**2 >= 56)

# Optimize
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)
    print('x6: ', x6.varValue)
else:
    print('No solution found')
```