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

## Step 2: Convert the natural language description into a symbolic representation
The symbolic representation of the variables is:
- $x_0$ : hours worked by Laura
- $x_1$ : hours worked by Bobby
- $x_2$ : hours worked by Jean

## Step 3: Define the objective function in symbolic notation
The objective function to minimize is $8x_0^2 + 6x_2^2 + x_0$.

## 4: List the constraints in symbolic notation
The constraints are:
- $6x_0 \geq 12$ (Laura's paperwork competence rating)
- $16x_1 \geq 294$ is incorrect based on the problem statement, it should be $6x_0 \geq 6$ for Laura's computer rating, $16x_1$ is correct for Bobby's computer rating but we need to account for all: 
- $6x_0$ (Laura's computer competence rating)
- $16x_1$ (Bobby's computer competence rating)
- $8x_2$ (Jean's computer competence rating)
- $7x_2$ (Jean's paperwork competence rating)
Correct constraints based on given data:
- $12x_0 \leq 169$ 
- $6x_0 \leq 294$ 
- $6x_1 \leq 169$ 
- $16x_1 \leq 294$ 
- $7x_2 \leq 169$ 
- $8x_2 \leq 294$
And the specific constraints:
- $6x_1^2 + 7x_2^2 \geq 25$
- $12x_0 + 6x_1 \geq 21$
- $12^2x_0^2 + 6^2x_1^2 + 7^2x_2^2 \geq 54$ is not directly stated, instead $12x_0 + 6x_1 + 7x_2 \geq 54$ 
- $6x_0 + 16x_1 \geq 40$
- $16x_1 + 8x_2 \geq 98$
- $6x_0 + 16x_1 + 8x_2 \geq 98$
- $4x_0^2 - 4x_2^2 \geq 0$

## 5: Create a JSON object representing the problem symbolically
```json
{
'sym_variables': [('x0', 'hours worked by Laura'), ('x1', 'hours worked by Bobby'), ('x2', 'hours worked by Jean')],
'objective_function': '8*x0^2 + 6*x2^2 + x0',
'constraints': [
'12*x0 <= 169',
'6*x0 <= 294',
'6*x1 <= 169',
'16*x1 <= 294',
'7*x2 <= 169',
'8*x2 <= 294',
'6*x1^2 + 7*x2^2 >= 25',
'12*x0 + 6*x1 >= 21',
'12*x0 + 6*x1 + 7*x2 >= 54',
'6*x0 + 16*x1 >= 40',
'16*x1 + 8*x2 >= 98',
'6*x0 + 16*x1 + 8*x2 >= 98',
'4*x0^2 - 4*x2^2 >= 0'
]
}
```

## 6: Write the Gurobi code to solve the optimization problem
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="hours_worked_by_Laura")
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="hours_worked_by_Bobby")
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="hours_worked_by_Jean")

    # Objective function
    model.setObjective(8*x0**2 + 6*x2**2 + x0, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(12*x0 <= 169)
    model.addConstr(6*x0 <= 294)
    model.addConstr(6*x1 <= 169)
    model.addConstr(16*x1 <= 294)
    model.addConstr(7*x2 <= 169)
    model.addConstr(8*x2 <= 294)
    model.addConstr(6*x1**2 + 7*x2**2 >= 25)
    model.addConstr(12*x0 + 6*x1 >= 21)
    model.addConstr(12*x0 + 6*x1 + 7*x2 >= 54)
    model.addConstr(6*x0 + 16*x1 >= 40)
    model.addConstr(16*x1 + 8*x2 >= 98)
    model.addConstr(6*x0 + 16*x1 + 8*x2 >= 98)
    model.addConstr(4*x0**2 - 4*x2**2 >= 0)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Laura: ", x0.varValue)
        print("Hours worked by Bobby: ", x1.varValue)
        print("Hours worked by Jean: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```