## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' corresponds to 'hours worked by John'
- 'x2' corresponds to 'hours worked by Hank'
- 'x3' corresponds to 'hours worked by Laura'
- 'x4' corresponds to 'hours worked by Jean'

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is: 
\[ 8x_1^2 + 8x_1x_2 + 9x_1x_3 + 2x_1x_4 + 7x_2x_3 + 6x_2x_4 + 3x_3^2 + 6x_4^2 + 3x_1 + 3x_2 + 2x_4 \]

## 3: Define the constraints in symbolic notation
The constraints are:
- $7x_1 \leq 221$
- $4x_2 \leq 221$
- $8x_3 \leq 221$
- $10x_4 \leq 221$
- $4x_2 + 10x_4 \geq 32$
- $7x_1 + 4x_2 \geq 28$
- $4x_2 + 8x_3 \geq 49$
- $8x_3 + 10x_4 \geq 41$
- $7x_1 + 4x_2 + 10x_4 \geq 55$
- $7x_1 + 4x_2 + 8x_3 + 10x_4 \geq 55$
- $-5x_1 + 5x_3 \geq 0$
- $4^2x_2^2 + 8^2x_3^2 + 10^2x_4^2 \leq 79$
- $7x_1 + 8x_3 + 10x_4 \leq 179$

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

## 5: Implement the objective function and constraints in Gurobi
```python
import gurobi

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_John", lb=0)
x2 = m.addVar(name="hours_worked_by_Hank", lb=0)
x3 = m.addVar(name="hours_worked_by_Laura", lb=0)
x4 = m.addVar(name="hours_worked_by_Jean", lb=0)

# Objective function
m.setObjective(8*x1**2 + 8*x1*x2 + 9*x1*x3 + 2*x1*x4 + 7*x2*x3 + 6*x2*x4 + 3*x3**2 + 6*x4**2 + 3*x1 + 3*x2 + 2*x4, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(7*x1 <= 221)
m.addConstr(4*x2 <= 221)
m.addConstr(8*x3 <= 221)
m.addConstr(10*x4 <= 221)
m.addConstr(4*x2 + 10*x4 >= 32)
m.addConstr(7*x1 + 4*x2 >= 28)
m.addConstr(4*x2 + 8*x3 >= 49)
m.addConstr(8*x3 + 10*x4 >= 41)
m.addConstr(7*x1 + 4*x2 + 10*x4 >= 55)
m.addConstr(7*x1 + 4*x2 + 8*x3 + 10*x4 >= 55)
m.addConstr(-5*x1 + 5*x3 >= 0)
m.addConstr(16*x2**2 + 64*x3**2 + 100*x4**2 <= 79)
m.addConstr(7*x1 + 8*x3 + 10*x4 <= 179)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by John: ", x1.varValue)
    print("Hours worked by Hank: ", x2.varValue)
    print("Hours worked by Laura: ", x3.varValue)
    print("Hours worked by Jean: ", x4.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by John'), 
        ('x2', 'hours worked by Hank'), 
        ('x3', 'hours worked by Laura'), 
        ('x4', 'hours worked by Jean')
    ], 
    'objective_function': '8*x1^2 + 8*x1*x2 + 9*x1*x3 + 2*x1*x4 + 7*x2*x3 + 6*x2*x4 + 3*x3^2 + 6*x4^2 + 3*x1 + 3*x2 + 2*x4', 
    'constraints': [
        '7*x1 <= 221', 
        '4*x2 <= 221', 
        '8*x3 <= 221', 
        '10*x4 <= 221', 
        '4*x2 + 10*x4 >= 32', 
        '7*x1 + 4*x2 >= 28', 
        '4*x2 + 8*x3 >= 49', 
        '8*x3 + 10*x4 >= 41', 
        '7*x1 + 4*x2 + 10*x4 >= 55', 
        '7*x1 + 4*x2 + 8*x3 + 10*x4 >= 55', 
        '-5*x1 + 5*x3 >= 0', 
        '16*x2^2 + 64*x3^2 + 100*x4^2 <= 79', 
        '7*x1 + 8*x3 + 10*x4 <= 179'
    ]
}
```