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

## Step 2: Convert the natural language objects into symbolic notation
The symbolic representation of the variables and their corresponding natural language objects are:
- $x_0$ : hours worked by Laura
- $x_1$ : hours worked by Peggy
- $x_2$ : hours worked by Bill

## 3: Define the objective function in symbolic notation
The objective function to maximize is $9x_0^2 + 2x_0x_1 + 3x_0x_2 + 9x_1^2 + 8x_1x_2 + 9x_2^2 + 3x_0 + 2x_1 + 9x_2$.

## 4: Define the constraints in symbolic notation
The constraints are:
- $6x_0 \leq 143$ (Laura's paperwork competence rating constraint, but it's an equality for her rating: $6x_0 = 6$)
- $7x_0 \leq 178$ (Laura's likelihood to quit index constraint)
- $3x_1 \leq 143$ (Peggy's paperwork competence rating constraint)
- $6x_1 \leq 178$ (Peggy's likelihood to quit index constraint)
- $2x_2 \leq 143$ (Bill's paperwork competence rating constraint)
- $14x_2 \leq 178$ (Bill's likelihood to quit index constraint)
- $7x_0 + 14x_2 \geq 20$ (combined likelihood to quit index from Laura and Bill)
- $7x_0 + 6x_1 + 14x_2 \geq 30$ (combined likelihood to quit index from all)
- $3x_1 + 2x_2 \leq 92$ (combined paperwork competence rating from Peggy and Bill)
- $6x_0 + 3x_1 \leq 100$ (combined paperwork competence rating from Laura and Peggy)
- $6x_0 + 3x_1 + 2x_2 \leq 100$ (combined paperwork competence rating from all)
- $6x_1 + 14x_2 \leq 121$ (combined likelihood to quit index from Peggy and Bill)
- $7x_0 + 6x_1 + 14x_2 \leq 76$ (combined likelihood to quit index from all, with equality for maximum: $7x_0 + 6x_1 + 14x_2 = 76$)
- $x_0, x_1 \geq 0$ (non-negativity constraints, as hours cannot be negative)
- $x_2$ is an integer (Bill's hours must be an integer)

## 5: Correct and refine constraints based on problem description
Given that Laura's, Peggy's, and Bill's individual ratings and indices are fixed:
- $6x_0 = 6$ implies $x_0 = 1$
- $7x_0 = 7$ is automatically satisfied if $x_0 = 1$
- $3x_1 = 3$ implies $x_1 = 1$
- $6x_1 = 6$ is automatically satisfied if $x_1 = 1$
- $2x_2 = 2$ implies $x_2 = 1$
- $14x_2 = 14$ is automatically satisfied if $x_2 = 1$

## 6: Formulate the problem in Gurobi
Given the corrections, we realize that $x_0$, $x_1$, and $x_2$ are actually determined directly by their respective constraints:
- $x_0 = 1$
- $x_1 = 1$
- $x_2 = 1$ and must be an integer.

The problem then simplifies significantly because the values of $x_0$, $x_1$, and $x_2$ are directly determined.

## 7: Write the Gurobi code
```python
import gurobi

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

# Define the variables
x0 = m.addVar(lb=0, name="hours_worked_by_Laura", vtype=gurobi.GRB.CONTINUOUS)
x1 = m.addVar(lb=0, name="hours_worked_by_Peggy", vtype=gurobi.GRB.CONTINUOUS)
x2 = m.addVar(lb=0, name="hours_worked_by_Bill", vtype=gurobi.GRB.INTEGER)

# Objective function
m.setObjective(9*x0**2 + 2*x0*x1 + 3*x0*x2 + 9*x1**2 + 8*x1*x2 + 9*x2**2 + 3*x0 + 2*x1 + 9*x2, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(6*x0 == 6)
m.addConstr(7*x0 <= 178)
m.addConstr(3*x1 == 3)
m.addConstr(6*x1 <= 178)
m.addConstr(2*x2 == 2)
m.addConstr(14*x2 <= 178)
m.addConstr(7*x0 + 14*x2 >= 20)
m.addConstr(7*x0 + 6*x1 + 14*x2 >= 30)
m.addConstr(3*x1 + 2*x2 <= 92)
m.addConstr(6*x0 + 3*x1 <= 100)
m.addConstr(6*x0 + 3*x1 + 2*x2 <= 100)
m.addConstr(6*x1 + 14*x2 <= 121)
m.addConstr(7*x0 + 6*x1 + 14*x2 <= 76)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 8: Symbolic Representation
```json
{
    'sym_variables': [('x0', 'hours worked by Laura'), ('x1', 'hours worked by Peggy'), ('x2', 'hours worked by Bill')],
    'objective_function': '9*x0^2 + 2*x0*x1 + 3*x0*x2 + 9*x1^2 + 8*x1*x2 + 9*x2^2 + 3*x0 + 2*x1 + 9*x2',
    'constraints': [
        '6*x0 = 6',
        '7*x0 <= 178',
        '3*x1 = 3',
        '6*x1 <= 178',
        '2*x2 = 2',
        '14*x2 <= 178',
        '7*x0 + 14*x2 >= 20',
        '7*x0 + 6*x1 + 14*x2 >= 30',
        '3*x1 + 2*x2 <= 92',
        '6*x0 + 3*x1 <= 100',
        '6*x0 + 3*x1 + 2*x2 <= 100',
        '6*x1 + 14*x2 <= 121',
        '7*x0 + 6*x1 + 14*x2 <= 76'
    ]
}
```