## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Ringo', 'hours worked by Paul', 'hours worked by Hank', 'hours worked by Peggy']. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to minimize is $5x_0 + 4x_1 + 4x_2 + 5x_3$.

## Step 3: List all the constraints
1. $17x_0 \leq 342$ and $7x_0 \leq 186$ are implicit in the variable definitions but we focus on given constraints.
2. $23x_1 \leq 342$ and $23x_1 \leq 186$ are also implicit.
3. $6x_2 \leq 342$ and $18x_2 \leq 186$ are implicit.
4. $17x_3 \leq 342$ and $19x_3 \leq 186$ are implicit.
5. $17x_0 + 23x_1 \geq 60$
6. $17x_0 + 6x_2 \geq 35$
7. $23x_1 + 6x_2 + 17x_3 \geq 77$
8. $17x_0 + 23x_1 + 6x_2 + 17x_3 \geq 77$
9. $7x_0 + 18x_2 \geq 19$
10. $23x_1 + 19x_3 \geq 24$
11. $7x_0 + 19x_3 \geq 36$
12. $7x_0 + 23x_1 + 19x_3 \geq 35$
13. $7x_0 + 23x_1 + 18x_2 \geq 35$
14. $7x_0 + 23x_1 + 18x_2 \geq 37$
15. $7x_0 + 23x_1 + 18x_2 + 19x_3 \geq 37$
16. $4x_1 - 8x_2 \geq 0$
17. $-8x_1 + 10x_3 \geq 0$
18. $-7x_0 + 9x_2 + 10x_3 \geq 0$
19. $17x_0 + 23x_1 + 17x_3 \leq 231$
20. $17x_0 + 6x_2 + 17x_3 \leq 104$
21. $23x_1 + 6x_2 + 17x_3 \leq 96$
22. $17x_0 + 23x_1 + 6x_2 \leq 276$
23. $7x_0 + 23x_1 \leq 58$
24. $23x_1 + 18x_2 \leq 118$
And $x_0, x_1, x_2, x_3$ are integers.

## 4: Convert the problem into Gurobi code
```python
import gurobi as gp

# Define the model
m = gp.Model()

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

# Define the objective function
m.setObjective(5 * x0 + 4 * x1 + 4 * x2 + 5 * x3, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(17 * x0 + 23 * x1 >= 60)
m.addConstr(17 * x0 + 6 * x2 >= 35)
m.addConstr(23 * x1 + 6 * x2 + 17 * x3 >= 77)
m.addConstr(17 * x0 + 23 * x1 + 6 * x2 + 17 * x3 >= 77)
m.addConstr(7 * x0 + 18 * x2 >= 19)
m.addConstr(23 * x1 + 19 * x3 >= 24)
m.addConstr(7 * x0 + 19 * x3 >= 36)
m.addConstr(7 * x0 + 23 * x1 + 19 * x3 >= 35)
m.addConstr(7 * x0 + 23 * x1 + 18 * x2 >= 35)
m.addConstr(7 * x0 + 23 * x1 + 18 * x2 >= 37)
m.addConstr(7 * x0 + 23 * x1 + 18 * x2 + 19 * x3 >= 37)
m.addConstr(4 * x1 - 8 * x2 >= 0)
m.addConstr(-8 * x1 + 10 * x3 >= 0)
m.addConstr(-7 * x0 + 9 * x2 + 10 * x3 >= 0)
m.addConstr(17 * x0 + 23 * x1 + 17 * x3 <= 231)
m.addConstr(17 * x0 + 6 * x2 + 17 * x3 <= 104)
m.addConstr(23 * x1 + 6 * x2 + 17 * x3 <= 96)
m.addConstr(17 * x0 + 23 * x1 + 6 * x2 <= 276)
m.addConstr(7 * x0 + 23 * x1 <= 58)
m.addConstr(23 * x1 + 18 * x2 <= 118)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Ringo: ", x0.varValue)
    print("Hours worked by Paul: ", x1.varValue)
    print("Hours worked by Hank: ", x2.varValue)
    print("Hours worked by Peggy: ", x3.varValue)
else:
    print("The problem is infeasible")
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Ringo'),
        ('x1', 'hours worked by Paul'),
        ('x2', 'hours worked by Hank'),
        ('x3', 'hours worked by Peggy')
    ],
    'objective_function': '5*x0 + 4*x1 + 4*x2 + 5*x3',
    'constraints': [
        '17*x0 + 23*x1 >= 60',
        '17*x0 + 6*x2 >= 35',
        '23*x1 + 6*x2 + 17*x3 >= 77',
        '17*x0 + 23*x1 + 6*x2 + 17*x3 >= 77',
        '7*x0 + 18*x2 >= 19',
        '23*x1 + 19*x3 >= 24',
        '7*x0 + 19*x3 >= 36',
        '7*x0 + 23*x1 + 19*x3 >= 35',
        '7*x0 + 23*x1 + 18*x2 >= 35',
        '7*x0 + 23*x1 + 18*x2 >= 37',
        '7*x0 + 23*x1 + 18*x2 + 19*x3 >= 37',
        '4*x1 - 8*x2 >= 0',
        '-8*x1 + 10*x3 >= 0',
        '-7*x0 + 9*x2 + 10*x3 >= 0',
        '17*x0 + 23*x1 + 17*x3 <= 231',
        '17*x0 + 6*x2 + 17*x3 <= 104',
        '23*x1 + 6*x2 + 17*x3 <= 96',
        '17*x0 + 23*x1 + 6*x2 <= 276',
        '7*x0 + 23*x1 <= 58',
        '23*x1 + 18*x2 <= 118'
    ]
}
```