## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by George', 'hours worked by Paul', 'hours worked by Peggy', and 'hours worked by Laura', 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 maximize is $4.45x_0 + 9.77x_1 + 6.93x_2 + 9.48x_3$.

## Step 3: Define the symbolic representation of the constraints
The constraints given are:
- $10x_0 = 10$ (George's likelihood to quit index)
- $10x_0 = 10$ (George's productivity rating)
- $8x_1 = 8$ (Paul's likelihood to quit index)
- $12x_1 = 12$ (Paul's productivity rating)
- $5x_2 = 5$ (Peggy's likelihood to quit index)
- $9x_2 = 9$ (Peggy's productivity rating)
- $12x_3 = 12$ (Laura's likelihood to quit index)
- $9x_3 = 9$ (Laura's productivity rating)
- $10x_0 + 9x_2 \geq 19$ (Total combined productivity rating from George and Peggy)
- $9x_2 + 9x_3 \geq 34$ (Total combined productivity rating from Peggy and Laura)
- $12x_1 + 9x_2 + 9x_3 \geq 42$ (Total combined productivity rating from Paul, Peggy, and Laura)
- $10x_0 + 9x_2 + 9x_3 \geq 42$ (Total combined productivity rating from George, Peggy, and Laura)
- $12x_1 + 9x_2 + 9x_3 \geq 24$ (Total combined productivity rating from Paul, Peggy, and Laura)
- $10x_0 + 9x_2 + 9x_3 \geq 24$ (Total combined productivity rating from George, Peggy, and Laura)
- $-8x_1 + 3x_3 \geq 0$ (Constraint involving Paul and Laura)
- $10x_0 + 12x_3 \leq 21$ (Total combined likelihood to quit index from George and Laura)
- $8x_1 + 5x_2 \leq 32$ (Total combined likelihood to quit index from Paul and Peggy)
- $10x_0 + 8x_1 \leq 40$ (Total combined likelihood to quit index from George and Paul)
- $10x_0 + 8x_1 + 5x_2 + 12x_3 \leq 40$ (Total combined likelihood to quit index from all)
- $12x_1 + 9x_3 \leq 113$ (Total combined productivity rating from Paul and Laura)
- $10x_0 + 9x_3 \leq 71$ (Total combined productivity rating from George and Laura)
- $12x_1 + 9x_2 \leq 142$ (Total combined productivity rating from Paul and Peggy)
- $10x_0 + 12x_1 + 9x_2 + 9x_3 \leq 142$ (Total combined productivity rating from all)

## 4: Correct and simplify constraints
Some constraints are equality constraints that can be directly used to fix variable values:
- $x_0 = 1$ (from $10x_0 = 10$)
- $x_1 = 1$ (from $12x_1 = 12$)
- $x_2 = 1$ (from $9x_2 = 9$)
- $x_3 = 1$ (from $9x_3 = 9$)

## 5: Realize that direct substitution simplifies the problem significantly
Substituting $x_0 = 1$, $x_1 = 1$, $x_2 = 1$, and $x_3 = 1$ into the objective function and constraints will simplify the problem.

## 6: Substitute fixed values into the objective function
The objective function becomes $4.45(1) + 9.77(1) + 6.93(1) + 9.48(1) = 4.45 + 9.77 + 6.93 + 9.48 = 30.63$.

## 7: Check if the fixed values satisfy all constraints
- $10(1) + 9(1) = 19 \geq 19$ (Satisfied)
- $9(1) + 9(1) = 18 \geq 34$ (Not satisfied)

## 8: Conclusion on feasibility
Since one of the constraints is not satisfied with the fixed values, the problem as stated with these equalities is infeasible.

## 9: Gurobi Code
Given the complexity and the specific request for Gurobi code, we need to model the problem without directly substituting the equality constraints.

```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by George
x1 = m.addVar(name="x1", lb=0, integrality=gp.GRB.INTEGER)  # hours worked by Paul
x2 = m.addVar(name="x2", lb=0)  # hours worked by Peggy
x3 = m.addVar(name="x3", lb=0)  # hours worked by Laura

# Objective function
m.setObjective(4.45*x0 + 9.77*x1 + 6.93*x2 + 9.48*x3, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(10*x0 <= 80, name="r0_George")
m.addConstr(10*x0 <= 169, name="r1_George")
m.addConstr(8*x1 <= 80, name="r0_Paul")
m.addConstr(12*x1 <= 169, name="r1_Paul")
m.addConstr(5*x2 <= 80, name="r0_Peggy")
m.addConstr(9*x2 <= 169, name="r1_Peggy")
m.addConstr(12*x3 <= 80, name="r0_Laura")
m.addConstr(9*x3 <= 169, name="r1_Laura")

m.addConstr(10*x0 + 9*x2 >= 19, name="prod_George_Peggy")
m.addConstr(9*x2 + 9*x3 >= 34, name="prod_Peggy_Laura")
m.addConstr(12*x1 + 9*x2 + 9*x3 >= 42, name="prod_Paul_Peggy_Laura")
m.addConstr(10*x0 + 9*x2 + 9*x3 >= 42, name="prod_George_Peggy_Laura")
m.addConstr(12*x1 + 9*x2 + 9*x3 >= 24, name="prod_Paul_Peggy_Laura_2")
m.addConstr(10*x0 + 9*x2 + 9*x3 >= 24, name="prod_George_Peggy_Laura_2")
m.addConstr(-8*x1 + 3*x3 >= 0, name="paul_laura_constraint")
m.addConstr(10*x0 + 12*x3 <= 21, name="quit_George_Laura")
m.addConstr(8*x1 + 5*x2 <= 32, name="quit_Paul_Peggy")
m.addConstr(10*x0 + 8*x1 <= 40, name="quit_George_Paul")
m.addConstr(10*x0 + 8*x1 + 5*x2 + 12*x3 <= 40, name="quit_all")
m.addConstr(12*x1 + 9*x3 <= 113, name="prod_Paul_Laura")
m.addConstr(10*x0 + 9*x3 <= 71, name="prod_George_Laura")
m.addConstr(12*x1 + 9*x2 <= 142, name="prod_Paul_Peggy")
m.addConstr(10*x0 + 12*x1 + 9*x2 + 9*x3 <= 142, name="prod_all")

# Solve the model
m.optimize()

if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found")
    print("Objective: ", m.objVal)
    print("x0: ", x0.varValue)
    print("x1: ", x1.varValue)
    print("x2: ", x2.varValue)
    print("x3: ", x3.varValue)
else:
    print("No optimal solution found")
```

```
{'sym_variables': [('x0', 'hours worked by George'), ('x1', 'hours worked by Paul'), ('x2', 'hours worked by Peggy'), ('x3', 'hours worked by Laura')], 'objective_function': '4.45*x0 + 9.77*x1 + 6.93*x2 + 9.48*x3', 'constraints': ['10*x0 <= 80', '10*x0 <= 169', '8*x1 <= 80', '12*x1 <= 169', '5*x2 <= 80', '9*x2 <= 169', '12*x3 <= 80', '9*x3 <= 169', '10*x0 + 9*x2 >= 19', '9*x2 + 9*x3 >= 34', '12*x1 + 9*x2 + 9*x3 >= 42', '10*x0 + 9*x2 + 9*x3 >= 42', '12*x1 + 9*x2 + 9*x3 >= 24', '10*x0 + 9*x2 + 9*x3 >= 24', '-8*x1 + 3*x3 >= 0', '10*x0 + 12*x3 <= 21', '8*x1 + 5*x2 <= 32', '10*x0 + 8*x1 <= 40', '10*x0 + 8*x1 + 5*x2 + 12*x3 <= 40', '12*x1 + 9*x3 <= 113', '10*x0 + 9*x3 <= 71', '12*x1 + 9*x2 <= 142', '10*x0 + 12*x1 + 9*x2 + 9*x3 <= 142']}
```