## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Jean', 'hours worked by Laura', and 'hours worked by Mary', 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 maximize is $4x_0 + 5x_1 + 4x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $19x_0 \geq 0$ (implicit, as $x_0$ is hours worked and cannot be negative, but we focus on given constraints)
- $12x_0 \geq 0$ (implicit, same reason)
- $8x_1 \geq 0$ (implicit)
- $14x_1 \geq 0$ (implicit)
- $16x_2 \geq 0$ (implicit)
- $2x_2 \geq 0$ (implicit)
- $19x_0 + 16x_2 \geq 22$
- $19x_0 + 8x_1 \geq 30$
- $8x_1 + 16x_2 \geq 27$
- $19x_0 + 8x_1 \leq 39$
- $8x_1 + 16x_2 \leq 91$
- $19x_0 + 8x_1 + 16x_2 \leq 91$
- $12x_0 + 2x_2 \leq 90$
- $12x_0 + 14x_1 \leq 56$
- $12x_0 + 14x_1 + 2x_2 \leq 56$

## 4: Convert the problem into a Gurobi model
We will use Gurobi to solve this linear programming problem.

## 5: Write the Gurobi code
```python
import gurobi as gp

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

# Define the variables
x0 = m.addVar(name="hours_worked_by_Jean", lb=0)  # hours worked by Jean
x1 = m.addVar(name="hours_worked_by_Laura", lb=0)  # hours worked by Laura
x2 = m.addVar(name="hours_worked_by_Mary", lb=0)  # hours worked by Mary

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

# Add constraints
m.addConstr(19*x0 + 16*x2 >= 22, name="work_quality_Jean_Mary")
m.addConstr(19*x0 + 8*x1 >= 30, name="work_quality_Jean_Laura")
m.addConstr(8*x1 + 16*x2 >= 27, name="work_quality_Laura_Mary")
m.addConstr(19*x0 + 8*x1 <= 39, name="work_quality_Jean_Laura_upper")
m.addConstr(8*x1 + 16*x2 <= 91, name="work_quality_Laura_Mary_upper")
m.addConstr(19*x0 + 8*x1 + 16*x2 <= 91, name="work_quality_total_upper")
m.addConstr(12*x0 + 2*x2 <= 90, name="cost_Jean_Mary")
m.addConstr(12*x0 + 14*x1 <= 56, name="cost_Jean_Laura")
m.addConstr(12*x0 + 14*x1 + 2*x2 <= 56, name="cost_total")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Jean: {x0.varValue}")
    print(f"Hours worked by Laura: {x1.varValue}")
    print(f"Hours worked by Mary: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Jean'), 
        ('x1', 'hours worked by Laura'), 
        ('x2', 'hours worked by Mary')
    ], 
    'objective_function': '4*x0 + 5*x1 + 4*x2', 
    'constraints': [
        '19*x0 + 16*x2 >= 22', 
        '19*x0 + 8*x1 >= 30', 
        '8*x1 + 16*x2 >= 27', 
        '19*x0 + 8*x1 <= 39', 
        '8*x1 + 16*x2 <= 91', 
        '19*x0 + 8*x1 + 16*x2 <= 91', 
        '12*x0 + 2*x2 <= 90', 
        '12*x0 + 14*x1 <= 56', 
        '12*x0 + 14*x1 + 2*x2 <= 56'
    ]
}
```