## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' corresponds to 'hours worked by Bill'
- 'x2' corresponds to 'hours worked by Ringo'
- 'x3' corresponds to 'hours worked by Mary'
- 'x4' corresponds to 'hours worked by John'

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is: 
\[ 3x_1^2 + 2x_1x_2 + 9x_1x_3 + 2x_1x_4 + 4x_2^2 + 7x_2x_3 + 5x_2x_4 + x_3x_4 + 2x_4^2 + 2x_1 + 4x_3 + 8x_4 \]

## Step 3: List the constraints in symbolic notation
The constraints are:
- $6x_1 \leq 203$ and $6x_1 \geq 0$ is not explicitly stated but implied by the context that ratings are used directly, so we consider $6x_1 = 6$ as a given condition.
- $20x_2 \leq 203$ and $20x_2 \geq 0$ is not explicitly stated but implied, so we consider $20x_2 = 20$ as a given condition.
- $5x_3 \leq 203$ and $5x_3 \geq 0$ is not explicitly stated but implied, so we consider $5x_3 = 5$ as a given condition.
- $3x_4 \leq 203$ and $3x_4 \geq 0$ is not explicitly stated but implied, so we consider $3x_4 = 3$ as a given condition.
- $6x_1 + 3x_4 \geq 19$
- $20x_2 + 3x_4 \geq 30$
- $20x_2 + 5x_3 \geq 39$
- $6x_1 + 20x_2 + 5x_3 + 3x_4 \geq 39$
- $-x_2^2 + 9x_3^2 \geq 0$
- $-9x_2 + 9x_4 \geq 0$
- $5x_1 - 2x_3 \geq 0$
- $6x_1 + 3x_4 \leq 201$
- $20x_2 + 3x_4 \leq 189$
- $20x_2 + 5x_3 + 3x_4 \leq 95$
- $6x_1 + 20x_2 + 5x_3 \leq 110$
- $6x_1 + 5x_3 + 3x_4 \leq 88$

## 4: Formulate the problem in Gurobi code
```python
import gurobi

def optimize_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="hours_worked_by_Bill", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    x2 = model.addVar(name="hours_worked_by_Ringo", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    x3 = model.addVar(name="hours_worked_by_Mary", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    x4 = model.addVar(name="hours_worked_by_John", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

    # Objective function
    model.setObjective(3*x1**2 + 2*x1*x2 + 9*x1*x3 + 2*x1*x4 + 4*x2**2 + 7*x2*x3 + 5*x2*x4 + x3*x4 + 2*x4**2 + 2*x1 + 4*x3 + 8*x4)

    # Constraints
    model.addConstr(6*x1 == 6, name="Bill_computer_competence")
    model.addConstr(20*x2 == 20, name="Ringo_computer_competence")
    model.addConstr(5*x3 == 5, name="Mary_computer_competence")
    model.addConstr(3*x4 == 3, name="John_computer_competence")

    model.addConstr(6*x1 + 3*x4 >= 19, name="Bill_John_total")
    model.addConstr(20*x2 + 3*x4 >= 30, name="Ringo_John_total")
    model.addConstr(20*x2 + 5*x3 >= 39, name="Ringo_Mary_total")
    model.addConstr(6*x1 + 20*x2 + 5*x3 + 3*x4 >= 39, name="All_total")

    model.addConstr(-x2**2 + 9*x3**2 >= 0, name="Ringo_Mary_constraint")
    model.addConstr(-9*x2 + 9*x4 >= 0, name="Ringo_John_constraint")
    model.addConstr(5*x1 - 2*x3 >= 0, name="Bill_Mary_constraint")

    model.addConstr(6*x1 + 3*x4 <= 201, name="Bill_John_limit")
    model.addConstr(20*x2 + 3*x4 <= 189, name="Ringo_John_limit")
    model.addConstr(20*x2 + 5*x3 + 3*x4 <= 95, name="Ringo_Mary_John_limit")
    model.addConstr(6*x1 + 20*x2 + 5*x3 <= 110, name="Bill_Ringo_Mary_limit")
    model.addConstr(6*x1 + 5*x3 + 3*x4 <= 88, name="Bill_Mary_John_limit")

    # Solve the problem
    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Bill: {x1.varValue}")
        print(f"Hours worked by Ringo: {x2.varValue}")
        print(f"Hours worked by Mary: {x3.varValue}")
        print(f"Hours worked by John: {x4.varValue}")
    else:
        print("No optimal solution found.")

optimize_problem()
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Bill'),
        ('x2', 'hours worked by Ringo'),
        ('x3', 'hours worked by Mary'),
        ('x4', 'hours worked by John')
    ],
    'objective_function': '3*x1^2 + 2*x1*x2 + 9*x1*x3 + 2*x1*x4 + 4*x2^2 + 7*x2*x3 + 5*x2*x4 + x3*x4 + 2*x4^2 + 2*x1 + 4*x3 + 8*x4',
    'constraints': [
        '6*x1 = 6',
        '20*x2 = 20',
        '5*x3 = 5',
        '3*x4 = 3',
        '6*x1 + 3*x4 >= 19',
        '20*x2 + 3*x4 >= 30',
        '20*x2 + 5*x3 >= 39',
        '6*x1 + 20*x2 + 5*x3 + 3*x4 >= 39',
        '-x2^2 + 9*x3^2 >= 0',
        '-9*x2 + 9*x4 >= 0',
        '5*x1 - 2*x3 >= 0',
        '6*x1 + 3*x4 <= 201',
        '20*x2 + 3*x4 <= 189',
        '20*x2 + 5*x3 + 3*x4 <= 95',
        '6*x1 + 20*x2 + 5*x3 <= 110',
        '6*x1 + 5*x3 + 3*x4 <= 88'
    ]
}
```