## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank', 'hours worked by Bill', 'hours worked by Jean', and 'hours worked by Dale', which can be represented symbolically as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $1.77x_0^2 + 8.58x_0x_1 + 2.9x_0x_2 + 9.51x_0x_3 + 5.01x_1^2 + 3.71x_1x_2 + 3.57x_1x_3 + 3.62x_2x_3 + 4.16x_3^2 + 7.54x_0 + 6.0x_1 + 7.17x_2 + 8.16x_3$.

## 3: List the constraints in symbolic notation
The constraints are:
- $2x_0 \leq 133$ (Implicitly, $x_0$ has an upper bound, but we focus on given constraints)
- $6x_0 \leq 104$
- $5x_1 \leq 133$
- $8x_1 \leq 104$
- $1x_2 \leq 133$
- $1x_2 \leq 104$
- $3x_3 \leq 133$
- $2x_3 \leq 104$
- $5x_1 + 3x_3 \geq 28$
- $5x_1 + 1x_2 \geq 24$
- $1x_2^2 + 3x_3^2 \geq 13$
- $2x_0^2 + 5x_1^2 \geq 23$
- $2x_0^2 + 1x_2^2 + 3x_3^2 \geq 19$
- $2x_0^2 + 5x_1^2 + 3x_3^2 \geq 19$
- $2x_0^2 + 1x_2^2 + 3x_3^2 \geq 28$
- $x_0 + x_1 + x_3 \geq 28$
- $x_0 + x_1 + x_2 + x_3 \geq 28$
- $8x_1 + 1x_2 \geq 26$
- $6x_0 + 8x_1 + 1x_2 + 2x_3 \geq 26$
- $-7x_1^2 + 1x_3^2 \geq 0$
- $5x_1 + 1x_2 \leq 118$
- $5x_1 + 3x_3 \leq 69$
- $8x_1^2 + 2x_3^2 \leq 62$
- $6x_0^2 + 2x_3^2 \leq 98$
- $6x_0 + 8x_1 + 2x_3 \leq 91$
- $x_0$ is an integer
- $x_1$ is an integer
- $x_2$ is not necessarily an integer
- $x_3$ is not necessarily an integer

## 4: Create a symbolic representation of the problem
```json
{
'sym_variables': [
    ('x0', 'hours worked by Hank'),
    ('x1', 'hours worked by Bill'),
    ('x2', 'hours worked by Jean'),
    ('x3', 'hours worked by Dale')
],
'objective_function': '1.77*x0^2 + 8.58*x0*x1 + 2.9*x0*x2 + 9.51*x0*x3 + 5.01*x1^2 + 3.71*x1*x2 + 3.57*x1*x3 + 3.62*x2*x3 + 4.16*x3^2 + 7.54*x0 + 6.0*x1 + 7.17*x2 + 8.16*x3',
'constraints': [
    'x0 >= 0',
    'x1 >= 0',
    'x2 >= 0',
    'x3 >= 0',
    '5*x1 + 3*x3 >= 28',
    '5*x1 + 1*x2 >= 24',
    '1*x2^2 + 3*x3^2 >= 13',
    '2*x0^2 + 5*x1^2 >= 23',
    '2*x0^2 + 1*x2^2 + 3*x3^2 >= 19',
    '2*x0^2 + 5*x1^2 + 3*x3^2 >= 19',
    '2*x0^2 + 1*x2^2 + 3*x3^2 >= 28',
    'x0 + x1 + x3 >= 28',
    'x0 + x1 + x2 + x3 >= 28',
    '8*x1 + 1*x2 >= 26',
    '6*x0 + 8*x1 + 1*x2 + 2*x3 >= 26',
    '-7*x1^2 + 1*x3^2 >= 0',
    '5*x1 + 1*x2 <= 118',
    '5*x1 + 3*x3 <= 69',
    '8*x1^2 + 2*x3^2 <= 62',
    '6*x0^2 + 2*x3^2 <= 98',
    '6*x0 + 8*x1 + 2*x3 <= 91'
]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi

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

# Define the variables
x0 = m.addVar(name='x0', vtype=gurobi.GRB.INTEGER)  # hours worked by Hank
x1 = m.addVar(name='x1', vtype=gurobi.GRB.INTEGER)  # hours worked by Bill
x2 = m.addVar(name='x2')  # hours worked by Jean
x3 = m.addVar(name='x3')  # hours worked by Dale

# Objective function
m.setObjective(1.77*x0**2 + 8.58*x0*x1 + 2.9*x0*x2 + 9.51*x0*x3 + 5.01*x1**2 + 3.71*x1*x2 + 3.57*x1*x3 + 3.62*x2*x3 + 4.16*x3**2 + 7.54*x0 + 6.0*x1 + 7.17*x2 + 8.16*x3, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(5*x1 + 3*x3 >= 28)
m.addConstr(5*x1 + 1*x2 >= 24)
m.addConstr(1*x2**2 + 3*x3**2 >= 13)
m.addConstr(2*x0**2 + 5*x1**2 >= 23)
m.addConstr(2*x0**2 + 1*x2**2 + 3*x3**2 >= 19)
m.addConstr(2*x0**2 + 5*x1**2 + 3*x3**2 >= 19)
m.addConstr(2*x0**2 + 1*x2**2 + 3*x3**2 >= 28)
m.addConstr(x0 + x1 + x3 >= 28)
m.addConstr(x0 + x1 + x2 + x3 >= 28)
m.addConstr(8*x1 + 1*x2 >= 26)
m.addConstr(6*x0 + 8*x1 + 1*x2 + 2*x3 >= 26)
m.addConstr(-7*x1**2 + 1*x3**2 >= 0)
m.addConstr(5*x1 + 1*x2 <= 118)
m.addConstr(5*x1 + 3*x3 <= 69)
m.addConstr(8*x1**2 + 2*x3**2 <= 62)
m.addConstr(6*x0**2 + 2*x3**2 <= 98)
m.addConstr(6*x0 + 8*x1 + 2*x3 <= 91)

# Optimize 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)
    print('x3: ', x3.varValue)
else:
    print('No solution found')
```