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

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $2.78x_0^2 + 3.17x_0x_2 + 1.44x_1^2 + 4.25x_2^2 + 1.63x_0 + 9.56x_2$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $3x_0 \leq 90$
- $1x_0 \leq 95$
- $13x_0 \leq 80$
- $10x_1 \leq 90$
- $6x_1 \leq 95$
- $10x_1 \leq 80$
- $6x_2 \leq 90$
- $12x_2 \leq 95$
- $4x_2 \leq 80$
- $3x_0^2 + 10x_1^2 \geq 23$
- $10x_1 + 6x_2 \geq 16$
- $3x_0 + 10x_1 + 6x_2 \geq 16$
- $6x_1 + 12x_2 \geq 15$
- $1x_0 + 12x_2 \geq 27$
- $1x_0 + 6x_1 + 12x_2 \geq 27$
- $13x_0^2 + 4x_2^2 \geq 21$
- $13x_0 + 10x_1 \geq 10$
- $13x_0 + 10x_1 + 4x_2 \geq 10$
- $2x_0 - 1x_2 \geq 0$
- $3x_1 - 10x_2 \geq 0$
- $10x_1^2 + 6x_2^2 \leq 32$
- $3x_0^2 + 10x_1^2 + 6x_2^2 \leq 35$
- $1x_0 + 6x_1 \leq 72$
- $1x_0^2 + 4x_2^2 \leq 95$
- $6x_1 + 12x_2 \leq 82$
- $x_0$ is an integer
- $x_1$ is continuous
- $x_2$ is an integer

## Step 4: Create the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Ringo'),
        ('x1', 'hours worked by Jean'),
        ('x2', 'hours worked by Peggy')
    ],
    'objective_function': '2.78*x0^2 + 3.17*x0*x2 + 1.44*x1^2 + 4.25*x2^2 + 1.63*x0 + 9.56*x2',
    'constraints': [
        '3*x0 <= 90',
        '1*x0 <= 95',
        '13*x0 <= 80',
        '10*x1 <= 90',
        '6*x1 <= 95',
        '10*x1 <= 80',
        '6*x2 <= 90',
        '12*x2 <= 95',
        '4*x2 <= 80',
        '3*x0^2 + 10*x1^2 >= 23',
        '10*x1 + 6*x2 >= 16',
        '3*x0 + 10*x1 + 6*x2 >= 16',
        '6*x1 + 12*x2 >= 15',
        '1*x0 + 12*x2 >= 27',
        '1*x0 + 6*x1 + 12*x2 >= 27',
        '13*x0^2 + 4*x2^2 >= 21',
        '13*x0 + 10*x1 >= 10',
        '13*x0 + 10*x1 + 4*x2 >= 10',
        '2*x0 - 1*x2 >= 0',
        '3*x1 - 10*x2 >= 0',
        '10*x1^2 + 6*x2^2 <= 32',
        '3*x0^2 + 10*x1^2 + 6*x2^2 <= 35',
        '1*x0 + 6*x1 <= 72',
        '1*x0^2 + 4*x2^2 <= 95',
        '6*x1 + 12*x2 <= 82'
    ]
}
```

## Step 5: Write the Gurobi code
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name='x0', vtype=gurobi.GRB.INTEGER)  # hours worked by Ringo
    x1 = model.addVar(name='x1')  # hours worked by Jean
    x2 = model.addVar(name='x2', vtype=gurobi.GRB.INTEGER)  # hours worked by Peggy

    # Define the objective function
    model.setObjective(2.78 * x0**2 + 3.17 * x0 * x2 + 1.44 * x1**2 + 4.25 * x2**2 + 1.63 * x0 + 9.56 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(3 * x0 <= 90)
    model.addConstr(1 * x0 <= 95)
    model.addConstr(13 * x0 <= 80)
    model.addConstr(10 * x1 <= 90)
    model.addConstr(6 * x1 <= 95)
    model.addConstr(10 * x1 <= 80)
    model.addConstr(6 * x2 <= 90)
    model.addConstr(12 * x2 <= 95)
    model.addConstr(4 * x2 <= 80)
    model.addConstr(3 * x0**2 + 10 * x1**2 >= 23)
    model.addConstr(10 * x1 + 6 * x2 >= 16)
    model.addConstr(3 * x0 + 10 * x1 + 6 * x2 >= 16)
    model.addConstr(6 * x1 + 12 * x2 >= 15)
    model.addConstr(1 * x0 + 12 * x2 >= 27)
    model.addConstr(1 * x0 + 6 * x1 + 12 * x2 >= 27)
    model.addConstr(13 * x0**2 + 4 * x2**2 >= 21)
    model.addConstr(13 * x0 + 10 * x1 >= 10)
    model.addConstr(13 * x0 + 10 * x1 + 4 * x2 >= 10)
    model.addConstr(2 * x0 - 1 * x2 >= 0)
    model.addConstr(3 * x1 - 10 * x2 >= 0)
    model.addConstr(10 * x1**2 + 6 * x2**2 <= 32)
    model.addConstr(3 * x0**2 + 10 * x1**2 + 6 * x2**2 <= 35)
    model.addConstr(1 * x0 + 6 * x1 <= 72)
    model.addConstr(1 * x0**2 + 4 * x2**2 <= 95)
    model.addConstr(6 * x1 + 12 * x2 <= 82)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('x0: ', x0.varValue)
        print('x1: ', x1.varValue)
        print('x2: ', x2.varValue)
    else:
        print('No solution found')

optimize_problem()
```