## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Peggy', 'hours worked by John', and 'hours worked by Laura', 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 Peggy
- $x_1$ : hours worked by John
- $x_2$ : hours worked by Laura

## 3: Define the objective function in symbolic notation
The objective function to maximize is: $3.21x_0^2 + 7.72x_0x_1 + 2.32x_0x_2 + 6.24x_1^2 + 7.47x_1x_2 + 3.51x_2^2 + 5.19x_0 + 4.76x_2$

## 4: List the constraints in symbolic notation
The constraints are:
- $10x_0 \leq 193$
- $8x_0 \leq 179$
- $28x_1 \leq 193$
- $28x_1 \leq 179$
- $29x_2 \leq 193$
- $20x_2 \leq 179$
- $28x_1 + 29x_2 \geq 57$
- $10x_0 + 28x_1 \geq 22$
- $8x_0 + 28x_1 \geq 45$
- $8x_0 + 20x_2 \geq 26$
- $28x_1 + 29x_2 \leq 154$
- $10x_0 + 28x_1 \leq 122$
- $10x_0 + 28x_1 + 29x_2 \leq 85$
- $28x_1 + 20x_2 \leq 67$
- $8x_0 + 28x_1 \leq 160$
- $8x_0 + 20x_2 \leq 76$
- $8x_0 + 28x_1 + 20x_2 \leq 121$
- $x_2$ is an integer

## 5: Create the Gurobi model and variables
We will now create a Gurobi model and define the variables.

## 6: Implement the objective function and constraints in Gurobi
```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Peggy
x1 = m.addVar(name="x1", lb=0)  # hours worked by John
x2 = m.addVar(name="x2", lb=0, integrality=gp.GRB.INTEGER)  # hours worked by Laura

# Define the objective function
m.setObjective(3.21*x0**2 + 7.72*x0*x1 + 2.32*x0*x2 + 6.24*x1**2 + 7.47*x1*x2 + 3.51*x2**2 + 5.19*x0 + 4.76*x2, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(10*x0 <= 193)
m.addConstr(8*x0 <= 179)
m.addConstr(28*x1 <= 193)
m.addConstr(28*x1 <= 179)
m.addConstr(29*x2 <= 193)
m.addConstr(20*x2 <= 179)
m.addConstr(28*x1 + 29*x2 >= 57)
m.addConstr(10*x0 + 28*x1 >= 22)
m.addConstr(8*x0 + 28*x1 >= 45)
m.addConstr(8*x0 + 20*x2 >= 26)
m.addConstr(28*x1 + 29*x2 <= 154)
m.addConstr(10*x0 + 28*x1 <= 122)
m.addConstr(10*x0 + 28*x1 + 29*x2 <= 85)
m.addConstr(28*x1 + 20*x2 <= 67)
m.addConstr(8*x0 + 28*x1 <= 160)
m.addConstr(8*x0 + 20*x2 <= 76)
m.addConstr(8*x0 + 28*x1 + 20*x2 <= 121)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print("Hours worked by Peggy: ", x0.varValue)
    print("Hours worked by John: ", x1.varValue)
    print("Hours worked by Laura: ", x2.varValue)
    print("Objective function value: ", m.objVal)
else:
    print("No optimal solution found.")
```

## 7: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Peggy'), 
        ('x1', 'hours worked by John'), 
        ('x2', 'hours worked by Laura')
    ], 
    'objective_function': '3.21*x0^2 + 7.72*x0*x1 + 2.32*x0*x2 + 6.24*x1^2 + 7.47*x1*x2 + 3.51*x2^2 + 5.19*x0 + 4.76*x2', 
    'constraints': [
        '10*x0 <= 193', 
        '8*x0 <= 179', 
        '28*x1 <= 193', 
        '28*x1 <= 179', 
        '29*x2 <= 193', 
        '20*x2 <= 179', 
        '28*x1 + 29*x2 >= 57', 
        '10*x0 + 28*x1 >= 22', 
        '8*x0 + 28*x1 >= 45', 
        '8*x0 + 20*x2 >= 26', 
        '28*x1 + 29*x2 <= 154', 
        '10*x0 + 28*x1 <= 122', 
        '10*x0 + 28*x1 + 29*x2 <= 85', 
        '28*x1 + 20*x2 <= 67', 
        '8*x0 + 28*x1 <= 160', 
        '8*x0 + 20*x2 <= 76', 
        '8*x0 + 28*x1 + 20*x2 <= 121'
    ]
}
```