## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bobby', 'hours worked by John', and 'hours worked by Paul', which can be represented as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is: $1 \cdot x_0 \cdot x_1 + 5 \cdot x_1^2 + 8 \cdot x_1 \cdot x_2 + 5 \cdot x_2^2 + 1 \cdot x_1 + 8 \cdot x_2$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $12x_0 = 12$ (Bobby's paperwork competence rating is 12)
- $20x_0 \leq 20 \cdot 1$ is not a constraint but $20x_0$ is given, similarly for others.
- $6x_0 = 6$ (Bobby's organization score is 6)
- $17x_0 = 17$ (Bobby's likelihood to quit index is 17)
- $14x_1 = 14$ (John's paperwork competence rating is 14)
- $7x_1 = 7$ (John has a dollar cost per hour of 7)
- $16x_1 = 16$ (John has an organization score of 16)
- $15x_1 = 15$ (John's likelihood to quit index is 15)
- $13x_2 = 13$ (Paul's paperwork competence rating is 13)
- $7x_2 = 7$ (Paul's dollar cost per hour is 7)
- $17x_2 = 17$ (Paul's organization score is 17)
- $7x_2 = 7$ (Paul's likelihood to quit index is 7)
- $6x_0 + 16x_1 + 17x_2 \geq 21$ (total combined organization score)
- $12x_0^2 + 14x_1^2 \leq 64$ (total combined paperwork competence rating from $x_0$ and $x_1$)
- $12x_0 + 13x_2 \leq 104$ (total combined paperwork competence rating from $x_0$ and $x_2$)
- $12x_0 + 14x_1 + 13x_2 \leq 104$ (total combined paperwork competence rating from all)
- $20x_0 + 7x_1 \leq 88$ (total combined dollar cost per hour from $x_0$ and $x_1$)
- $20x_0 + 7x_2 \leq 62$ (total combined dollar cost per hour from $x_0$ and $x_2$)
- $20x_0 + 7x_1 + 7x_2 \leq 62$ (total combined dollar cost per hour from all)
- $6x_0 + 17x_2 \leq 69$ (total combined organization score from $x_0$ and $x_2$)
- $6x_0 + 16x_1 \leq 39$ (total combined organization score from $x_0$ and $x_1$)
- $6x_0 + 16x_1 + 17x_2 \leq 98$ (total combined organization score from all)
- $15x_1 + 7x_2 \leq 109$ (total combined likelihood to quit index from $x_1$ and $x_2$)
- $17x_0 + 15x_1 \leq 73$ (total combined likelihood to quit index from $x_0$ and $x_1$)
- $17x_0 + 15x_1 + 7x_2 \leq 73$ (total combined likelihood to quit index from all)
- $x_0 \in \mathbb{Z}$ (non-fractional number of hours worked by Bobby)
- $x_1 \in \mathbb{Z}$ (non-fractional number of hours worked by John)
- $x_2 \in \mathbb{Z}$ (non-fractional number of hours worked by Paul)

## 4: Convert the problem into Gurobi code
```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="x0", vtype=gp.GRB.INTEGER)  # hours worked by Bobby
x1 = m.addVar(name="x1", vtype=gp.GRB.INTEGER)  # hours worked by John
x2 = m.addVar(name="x2", vtype=gp.GRB.INTEGER)  # hours worked by Paul

# Objective function
m.setObjective(x0 * x1 + 5 * x1 ** 2 + 8 * x1 * x2 + 5 * x2 ** 2 + x1 + 8 * x2, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(12 * x0 == 12, name="bobby_paperwork")
m.addConstr(6 * x0 == 6, name="bobby_organization")
m.addConstr(17 * x0 == 17, name="bobby_likelihood")

m.addConstr(14 * x1 == 14, name="john_paperwork")
m.addConstr(7 * x1 == 7, name="john_cost")
m.addConstr(16 * x1 == 16, name="john_organization")
m.addConstr(15 * x1 == 15, name="john_likelihood")

m.addConstr(13 * x2 == 13, name="paul_paperwork")
m.addConstr(7 * x2 == 7, name="paul_cost")
m.addConstr(17 * x2 == 17, name="paul_organization")
m.addConstr(7 * x2 == 7, name="paul_likelihood")

m.addConstr(6 * x0 + 16 * x1 + 17 * x2 >= 21, name="total_organization")
m.addConstr(12 * x0 ** 2 + 14 * x1 ** 2 <= 64, name="paperwork_competence")
m.addConstr(12 * x0 + 13 * x2 <= 104, name="paperwork_competence_bobby_paul")
m.addConstr(12 * x0 + 14 * x1 + 13 * x2 <= 104, name="paperwork_competence_all")
m.addConstr(20 * x0 + 7 * x1 <= 88, name="dollar_cost_bobby_john")
m.addConstr(20 * x0 + 7 * x2 <= 62, name="dollar_cost_bobby_paul")
m.addConstr(20 * x0 + 7 * x1 + 7 * x2 <= 62, name="dollar_cost_all")
m.addConstr(6 * x0 + 17 * x2 <= 69, name="organization_bobby_paul")
m.addConstr(6 * x0 + 16 * x1 <= 39, name="organization_bobby_john")
m.addConstr(6 * x0 + 16 * x1 + 17 * x2 <= 98, name="organization_all")
m.addConstr(15 * x1 + 7 * x2 <= 109, name="likelihood_john_paul")
m.addConstr(17 * x0 + 15 * x1 <= 73, name="likelihood_bobby_john")
m.addConstr(17 * x0 + 15 * x1 + 7 * x2 <= 73, name="likelihood_all")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Bobby: ", x0.varValue)
    print("Hours worked by John: ", x1.varValue)
    print("Hours worked by Paul: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'hours worked by Bobby'), ('x1', 'hours worked by John'), ('x2', 'hours worked by Paul')],
    'objective_function': 'x0 * x1 + 5 * x1^2 + 8 * x1 * x2 + 5 * x2^2 + x1 + 8 * x2',
    'constraints': [
        '12 * x0 = 12',
        '6 * x0 = 6',
        '17 * x0 = 17',
        '14 * x1 = 14',
        '7 * x1 = 7',
        '16 * x1 = 16',
        '15 * x1 = 15',
        '13 * x2 = 13',
        '7 * x2 = 7',
        '17 * x2 = 17',
        '7 * x2 = 7',
        '6 * x0 + 16 * x1 + 17 * x2 >= 21',
        '12 * x0^2 + 14 * x1^2 <= 64',
        '12 * x0 + 13 * x2 <= 104',
        '12 * x0 + 14 * x1 + 13 * x2 <= 104',
        '20 * x0 + 7 * x1 <= 88',
        '20 * x0 + 7 * x2 <= 62',
        '20 * x0 + 7 * x1 + 7 * x2 <= 62',
        '6 * x0 + 17 * x2 <= 69',
        '6 * x0 + 16 * x1 <= 39',
        '6 * x0 + 16 * x1 + 17 * x2 <= 98',
        '15 * x1 + 7 * x2 <= 109',
        '17 * x0 + 15 * x1 <= 73',
        '17 * x0 + 15 * x1 + 7 * x2 <= 73',
        'x0 \\in \\mathbb{Z}',
        'x1 \\in \\mathbb{Z}',
        'x2 \\in \\mathbb{Z}'
    ]
}
```