## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Dale', 'hours worked by Peggy', 'hours worked by Paul', 'hours worked by John'] which can be represented symbolically as 'x0', 'x1', 'x2', 'x3' respectively.

## Step 2: Convert the natural language description into a symbolic representation
The symbolic representation of the variables is:
- 'x0' for 'hours worked by Dale'
- 'x1' for 'hours worked by Peggy'
- 'x2' for 'hours worked by Paul'
- 'x3' for 'hours worked by John'

## 3: Define the objective function in symbolic notation
The objective function to minimize is: 
\[ 1 \cdot x0 \cdot x1 + 1 \cdot x0 \cdot x2 + 3 \cdot x0 \cdot x3 + 4 \cdot x1^2 + 8 \cdot x1 \cdot x2 + 6 \cdot x1 \cdot x3 + 7 \cdot x2 \cdot x3 + 1 \cdot x3^2 + 4 \cdot x0 + 8 \cdot x1 + 2 \cdot x2 + 2 \cdot x3 \]

## 4: List the constraints in symbolic notation
The constraints are:
- \( 5 \cdot x0 \leq 111 \)
- \( 7 \cdot x0 \leq 127 \)
- \( 8 \cdot x1 \leq 111 \)
- \( 2 \cdot x1 \leq 127 \)
- \( 11 \cdot x2 \leq 111 \)
- \( 2 \cdot x2 \leq 127 \)
- \( 4 \cdot x3 \leq 111 \)
- \( 9 \cdot x3 \leq 127 \)
- \( 11 \cdot x2 + 4 \cdot x3 \geq 25 \)
- \( 5^2 \cdot x0^2 + 11^2 \cdot x2^2 \geq 17 \)
- \( 8 \cdot x1 + 4 \cdot x3 \geq 25 \)
- \( 5 \cdot x0 + 4 \cdot x3 \geq 10 \)
- \( 5 \cdot x0 + 8 \cdot x1 + 11 \cdot x2 \geq 24 \)
- \( 5 \cdot x0 + 8 \cdot x1 + 11 \cdot x2 + 4 \cdot x3 \geq 24 \)
- \( 2 \cdot x1 + 9 \cdot x3 \geq 31 \)
- \( 7 \cdot x0 + 9 \cdot x3 \geq 16 \)
- \( 7 \cdot x0 + 2 \cdot x1 + 2 \cdot x2 \geq 31 \)
- \( 7 \cdot x0 + 2 \cdot x1 + 2 \cdot x2 + 9 \cdot x3 \geq 31 \)
- \( 2 \cdot x1 + 2 \cdot x2 \leq 80 \)
- \( 7 \cdot x0 + 9 \cdot x3 \leq 67 \)
- \( 7^2 \cdot x0^2 + 2^2 \cdot x2^2 + 9^2 \cdot x3^2 \leq 55 \)
- \( 7 \cdot x0 + 2 \cdot x1 + 2 \cdot x2 \leq 53 \)
- \( 2 \cdot x1 + 2 \cdot x2 + 9 \cdot x3 \leq 88 \)
- \( x0 \in \mathbb{Z} \)
- \( x1 \in \mathbb{Z} \)
- \( x2 \in \mathbb{Z} \)
- \( x3 \in \mathbb{Z} \)

## 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
Below is the Python code for the Gurobi implementation.

```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="x0", vtype=gp.GRB.INTEGER)  # hours worked by Dale
x1 = m.addVar(name="x1", vtype=gp.GRB.INTEGER)  # hours worked by Peggy
x2 = m.addVar(name="x2", vtype=gp.GRB.INTEGER)  # hours worked by Paul
x3 = m.addVar(name="x3", vtype=gp.GRB.INTEGER)  # hours worked by John

# Objective function
m.setObjective(1 * x0 * x1 + 1 * x0 * x2 + 3 * x0 * x3 + 4 * x1**2 + 8 * x1 * x2 + 6 * x1 * x3 + 7 * x2 * x3 + 1 * x3**2 + 4 * x0 + 8 * x1 + 2 * x2 + 2 * x3, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(5 * x0 <= 111)
m.addConstr(7 * x0 <= 127)
m.addConstr(8 * x1 <= 111)
m.addConstr(2 * x1 <= 127)
m.addConstr(11 * x2 <= 111)
m.addConstr(2 * x2 <= 127)
m.addConstr(4 * x3 <= 111)
m.addConstr(9 * x3 <= 127)
m.addConstr(11 * x2 + 4 * x3 >= 25)
m.addConstr(25 * x0**2 + 121 * x2**2 >= 17)
m.addConstr(8 * x1 + 4 * x3 >= 25)
m.addConstr(5 * x0 + 4 * x3 >= 10)
m.addConstr(5 * x0 + 8 * x1 + 11 * x2 >= 24)
m.addConstr(5 * x0 + 8 * x1 + 11 * x2 + 4 * x3 >= 24)
m.addConstr(2 * x1 + 9 * x3 >= 31)
m.addConstr(7 * x0 + 9 * x3 >= 16)
m.addConstr(7 * x0 + 2 * x1 + 2 * x2 >= 31)
m.addConstr(7 * x0 + 2 * x1 + 2 * x2 + 9 * x3 >= 31)
m.addConstr(2 * x1 + 2 * x2 <= 80)
m.addConstr(7 * x0 + 9 * x3 <= 67)
m.addConstr(49 * x0**2 + 4 * x2**2 + 81 * x3**2 <= 55)
m.addConstr(7 * x0 + 2 * x1 + 2 * x2 <= 53)
m.addConstr(2 * x1 + 2 * x2 + 9 * x3 <= 88)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
else:
    print("The model is infeasible")
```

## 7: Symbolic Representation
The symbolic representation of the problem is as follows:

```json
{
    'sym_variables': [
        ('x0', 'hours worked by Dale'),
        ('x1', 'hours worked by Peggy'),
        ('x2', 'hours worked by Paul'),
        ('x3', 'hours worked by John')
    ],
    'objective_function': '1 * x0 * x1 + 1 * x0 * x2 + 3 * x0 * x3 + 4 * x1^2 + 8 * x1 * x2 + 6 * x1 * x3 + 7 * x2 * x3 + 1 * x3^2 + 4 * x0 + 8 * x1 + 2 * x2 + 2 * x3',
    'constraints': [
        '5 * x0 <= 111',
        '7 * x0 <= 127',
        '8 * x1 <= 111',
        '2 * x1 <= 127',
        '11 * x2 <= 111',
        '2 * x2 <= 127',
        '4 * x3 <= 111',
        '9 * x3 <= 127',
        '11 * x2 + 4 * x3 >= 25',
        '25 * x0^2 + 121 * x2^2 >= 17',
        '8 * x1 + 4 * x3 >= 25',
        '5 * x0 + 4 * x3 >= 10',
        '5 * x0 + 8 * x1 + 11 * x2 >= 24',
        '5 * x0 + 8 * x1 + 11 * x2 + 4 * x3 >= 24',
        '2 * x1 + 9 * x3 >= 31',
        '7 * x0 + 9 * x3 >= 16',
        '7 * x0 + 2 * x1 + 2 * x2 >= 31',
        '7 * x0 + 2 * x1 + 2 * x2 + 9 * x3 >= 31',
        '2 * x1 + 2 * x2 <= 80',
        '7 * x0 + 9 * x3 <= 67',
        '49 * x0^2 + 4 * x2^2 + 81 * x3^2 <= 55',
        '7 * x0 + 2 * x1 + 2 * x2 <= 53',
        '2 * x1 + 2 * x2 + 9 * x3 <= 88',
        'x0 ∈ ℤ',
        'x1 ∈ ℤ',
        'x2 ∈ ℤ',
        'x3 ∈ ℤ'
    ]
}
```