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

## Step 2: Convert the given problem into a symbolic representation
The objective function to minimize is $3 \cdot x_0 \cdot x_1 + 6 \cdot x_0 + 2 \cdot x_1$.

## 3: List the constraints
1. $1.4x_1 + 3.79x_2 \geq 15$
2. $15.55^2x_0^2 + 1.4^2x_1^2 \geq 12$
3. $15.55^2x_0^2 + 1.4^2x_1^2 + 3.79^2x_2^2 \geq 13$
4. $15.55x_0 + 1.4x_1 + 3.79x_2 \geq 13$
5. $11.52x_0 + 8.25x_1 \geq 21$
6. $11.52x_0 + 8.25x_1 + 14.56x_2 \geq 21$
7. $8.96x_0 + 10.83x_2 \geq 34$
8. $9.37x_1 + 10.83x_2 \geq 15$
9. $8.96x_0 + 9.37x_1 + 10.83x_2 \geq 35$
10. $-4x_1 + 6x_2 \geq 0$
11. $-4x_0^2 + 4x_1^2 \geq 0$
12. $15.55x_0 + 1.4x_1 \leq 58$
13. $11.52x_0 + 8.25x_1 \leq 32$
14. $8.96x_0 + 10.83x_2 \leq 125$
15. $9.37^2x_1^2 + 10.83^2x_2^2 \leq 63$

## 4: Define the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Laura'),
        ('x1', 'hours worked by Ringo'),
        ('x2', 'hours worked by Jean')
    ],
    'objective_function': '3 * x0 * x1 + 6 * x0 + 2 * x1',
    'constraints': [
        '1.4 * x1 + 3.79 * x2 >= 15',
        '15.55^2 * x0^2 + 1.4^2 * x1^2 >= 12',
        '15.55^2 * x0^2 + 1.4^2 * x1^2 + 3.79^2 * x2^2 >= 13',
        '15.55 * x0 + 1.4 * x1 + 3.79 * x2 >= 13',
        '11.52 * x0 + 8.25 * x1 >= 21',
        '11.52 * x0 + 8.25 * x1 + 14.56 * x2 >= 21',
        '8.96 * x0 + 10.83 * x2 >= 34',
        '9.37 * x1 + 10.83 * x2 >= 15',
        '8.96 * x0 + 9.37 * x1 + 10.83 * x2 >= 35',
        '-4 * x1 + 6 * x2 >= 0',
        '-4 * x0^2 + 4 * x1^2 >= 0',
        '15.55 * x0 + 1.4 * x1 <= 58',
        '11.52 * x0 + 8.25 * x1 <= 32',
        '8.96 * x0 + 10.83 * x2 <= 125',
        '9.37^2 * x1^2 + 10.83^2 * x2^2 <= 63'
    ]
}
```

## 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

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

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Laura
x1 = m.addVar(name="x1", lb=0)  # hours worked by Ringo
x2 = m.addVar(name="x2", lb=0, integrality=1)  # hours worked by Jean

# Objective function
m.setObjective(3 * x0 * x1 + 6 * x0 + 2 * x1, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(1.4 * x1 + 3.79 * x2 >= 15)
m.addConstr(15.55**2 * x0**2 + 1.4**2 * x1**2 >= 12)
m.addConstr(15.55**2 * x0**2 + 1.4**2 * x1**2 + 3.79**2 * x2**2 >= 13)
m.addConstr(15.55 * x0 + 1.4 * x1 + 3.79 * x2 >= 13)
m.addConstr(11.52 * x0 + 8.25 * x1 >= 21)
m.addConstr(11.52 * x0 + 8.25 * x1 + 14.56 * x2 >= 21)
m.addConstr(8.96 * x0 + 10.83 * x2 >= 34)
m.addConstr(9.37 * x1 + 10.83 * x2 >= 15)
m.addConstr(8.96 * x0 + 9.37 * x1 + 10.83 * x2 >= 35)
m.addConstr(-4 * x1 + 6 * x2 >= 0)
m.addConstr(-4 * x0**2 + 4 * x1**2 >= 0)
m.addConstr(15.55 * x0 + 1.4 * x1 <= 58)
m.addConstr(11.52 * x0 + 8.25 * x1 <= 32)
m.addConstr(8.96 * x0 + 10.83 * x2 <= 125)
m.addConstr(9.37**2 * x1**2 + 10.83**2 * x2**2 <= 63)

# 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")
```