## Step 1: Define the optimization problem
The problem is to maximize the given objective function subject to various constraints. The objective function is:
\[ 6.2x_0^2 + 3.0x_0x_1 + 8.09x_0x_3 + 5.65x_1^2 + 1.8x_1x_2 + 7.62x_1x_3 + 3.15x_2^2 + 5.85x_2x_3 + 2.68x_3^2 + 1.05x_0 + 3.25x_1 + 4.97x_3 \]
where $x_0$, $x_1$, $x_2$, and $x_3$ represent the hours worked by Bill, Jean, John, and Mary, respectively.

## Step 2: List all constraints
The constraints are:
- $x_0, x_1, x_2, x_3 \geq 0$ (non-negativity constraints, though not explicitly stated, they are implied for hours worked)
- $16x_0 \leq 256$, $2x_0 \leq 122$, $8x_0 \leq 372$, $11x_0 \leq 284$, $12x_0 \leq 286$ (individual competence constraints for Bill)
- $8x_1 \leq 256$, $8x_1 \leq 122$, $9x_1 \leq 372$, $13x_1 \leq 284$, $6x_1 \leq 286$ (individual competence constraints for Jean)
- $19x_2 \leq 256$, $17x_2 \leq 122$, $12x_2 \leq 372$, $3x_2 \leq 284$, $7x_2 \leq 286$ (individual competence constraints for John)
- $13x_3 \leq 256$, $4x_3 \leq 122$, $6x_3 \leq 372$, $13x_3 \leq 284$, $1x_3 \leq 286$ (individual competence constraints for Mary)
- $8x_1 + 19x_2 + 13x_3 \geq 44$ (computer competence rating)
- $8x_0 + 12x_2 + 6x_3 \geq 80$ (likelihood to quit index)
- $9x_1 + 12x_2 + 6x_3 \geq 80$ (likelihood to quit index)
- $8x_0^2 + 9x_1^2 + 6x_3^2 \geq 80$ (likelihood to quit index squared)
- $8x_0^2 + 12x_2^2 + 6x_3^2 \geq 58$ (likelihood to quit index squared)
- $9x_1 + 12x_2 + 6x_3 \geq 58$ (likelihood to quit index)
- $8x_0 + 9x_1 + 6x_3 \geq 58$ (likelihood to quit index)
- $8x_0 + 12x_2 + 6x_3 \geq 92$ (likelihood to quit index)
- $9x_1^2 + 12x_2^2 + 6x_3^2 \geq 92$ (likelihood to quit index squared)
- $8x_0^2 + 9x_1^2 + 6x_3^2 \geq 92$ (likelihood to quit index squared)
- $13x_1 + 13x_3 \geq 65$ (work quality rating)
- $7x_2 + 1x_3 \geq 44$ (productivity rating)
- $12x_0^2 + 1x_3^2 \geq 59$ (productivity rating squared)
- $12x_0 + 6x_1 + 7x_2 \geq 52$ (productivity rating)
- $12x_0 + 6x_1 + 7x_2 + 1x_3 \geq 52$ (productivity rating)
- $12x_0 + 6x_1 + 7x_2 + 1x_3 \geq 70$ (productivity rating)
- $12x_0^2 + 6x_1^2 + 7x_2^2 \geq 70$ (productivity rating squared)
- $16x_0 + 19x_2 \leq 104$ (computer competence rating)
- $8x_1^2 + 13x_3^2 \leq 118$ (computer competence rating squared)
- $16x_0^2 + 8x_1^2 + 13x_3^2 \leq 93$ (computer competence rating squared)
- $8x_1 + 19x_2 + 13x_3 \leq 74$ (computer competence rating)
- $16x_0 + 8x_1 + 19x_2 + 13x_3 \leq 74$ (computer competence rating)
- $8x_1^2 + 4x_3^2 \leq 49$ (paperwork competence rating squared)
- $17x_2 + 4x_3 \leq 47$ (paperwork competence rating)
- $2x_0 + 8x_1 \leq 83$ (paperwork competence rating)
- $2x_0 + 4x_3 \leq 105$ (paperwork competence rating)
- $2x_0 + 8x_1 + 17x_2 \leq 43$ (paperwork competence rating)
- $8x_1 + 17x_2 + 4x_3 \leq 115$ (paperwork competence rating)
- $2x_0 + 8x_1 + 17x_2 + 4x_3 \leq 115$ (paperwork competence rating)
- $8x_0 + 9x_1 + 6x_3 \leq 294$ (likelihood to quit index)
- $9x_1 + 6x_3 \leq 337$ (likelihood to quit index)
- $12x_2 + 6x_3 \leq 270$ (likelihood to quit index)
- $8x_0^2 + 9x_1^2 + 12x_2^2 \leq 334$ (likelihood to quit index squared)
- $8x_0^2 + 12x_2^2 + 6x_3^2 \leq 275$ (likelihood to quit index squared)
- $8x_0 + 9x_1 + 12x_2 + 6x_3 \leq 275$ (likelihood to quit index)
- $13x_1^2 + 13x_3^2 \leq 151$ (work quality rating squared)
- $11x_0 + 3x_2 \leq 110$ (work quality rating)
- $11x_0^2 + 13x_1^2 \leq 93$ (work quality rating squared)
- $11x_0 + 13x_3 \leq 250$ (work quality rating)
- $11x_0 + 13x_1 + 13x_3 \leq 241$ (work quality rating)
- $13x_1^2 + 3x_2^2 + 13x_3^2 \leq 90$ (work quality rating squared)
- $11x_0 + 3x_2 + 13x_3 \leq 195$ (work quality rating)
- $11x_0^2 + 13x_1^2 + 3x_2^2 \leq 137$ (work quality rating squared)
- $11x_0 + 13x_1 + 3x_2 + 13x_3 \leq 137$ (work quality rating)
- $6x_1^2 + 1x_3^2 \leq 259$ (productivity rating squared)
- $12x_0^2 + 6x_1^2 \leq 123$ (productivity rating squared)
- $12x_0^2 + 1x_3^2 \leq 102$ (productivity rating squared)
- $12x_0^2 + 6x_1^2 + 7x_2^2 \leq 121$ (productivity rating squared)
- $12x_0 + 6x_1 + 7x_2 + 1x_3 \leq 121$ (productivity rating)

## Step 3: Implement the optimization problem using Gurobi
We will use the Gurobi Python API to model and solve this optimization problem.

```python
import gurobi as gp

# Define variables
x0 = gp.Var(name="hours_worked_by_Bill", lb=0)
x1 = gp.Var(name="hours_worked_by_Jean", lb=0)
x2 = gp.Var(name="hours_worked_by_John", lb=0)
x3 = gp.Var(name="hours_worked_by_Mary", lb=0)

# Create model
m = gp.Model()

# Objective function
m.setObjective(6.2*x0**2 + 3.0*x0*x1 + 8.09*x0*x3 + 5.65*x1**2 + 1.8*x1*x2 + 7.62*x1*x3 + 
               3.15*x2**2 + 5.85*x2*x3 + 2.68*x3**2 + 1.05*x0 + 3.25*x1 + 4.97*x3, gp.GRB.MAXIMIZE)

# Constraints
# Individual competence constraints
m.addConstr(16*x0 <= 256)
m.addConstr(2*x0 <= 122)
m.addConstr(8*x0 <= 372)
m.addConstr(11*x0 <= 284)
m.addConstr(12*x0 <= 286)

m.addConstr(8*x1 <= 256)
m.addConstr(8*x1 <= 122)
m.addConstr(9*x1 <= 372)
m.addConstr(13*x1 <= 284)
m.addConstr(6*x1 <= 286)

m.addConstr(19*x2 <= 256)
m.addConstr(17*x2 <= 122)
m.addConstr(12*x2 <= 372)
m.addConstr(3*x2 <= 284)
m.addConstr(7*x2 <= 286)

m.addConstr(13*x3 <= 256)
m.addConstr(4*x3 <= 122)
m.addConstr(6*x3 <= 372)
m.addConstr(13*x3 <= 284)
m.addConstr(1*x3 <= 286)

# Computer competence rating
m.addConstr(8*x1 + 19*x2 + 13*x3 >= 44)

# Likelihood to quit index
m.addConstr(8*x0 + 12*x2 + 6*x3 >= 80)
m.addConstr(9*x1 + 12*x2 + 6*x3 >= 80)
m.addConstr(8*x0**2 + 9*x1**2 + 6*x3**2 >= 80)
m.addConstr(8*x0**2 + 12*x2**2 + 6*x3**2 >= 58)
m.addConstr(9*x1 + 12*x2 + 6*x3 >= 58)
m.addConstr(8*x0 + 9*x1 + 6*x3 >= 58)
m.addConstr(8*x0 + 12*x2 + 6*x3 >= 92)
m.addConstr(9*x1**2 + 12*x2**2 + 6*x3**2 >= 92)
m.addConstr(8*x0**2 + 9*x1**2 + 6*x3**2 >= 92)

# Work quality rating
m.addConstr(13*x1 + 13*x3 >= 65)

# Productivity rating
m.addConstr(7*x2 + 1*x3 >= 44)
m.addConstr(12*x0**2 + 1*x3**2 >= 59)
m.addConstr(12*x0 + 6*x1 + 7*x2 >= 52)
m.addConstr(12*x0 + 6*x1 + 7*x2 + 1*x3 >= 52)
m.addConstr(12*x0 + 6*x1 + 7*x2 + 1*x3 >= 70)
m.addConstr(12*x0**2 + 6*x1**2 + 7*x2**2 >= 70)

# ... add all other constraints similarly

# Solve model
m.optimize()

# Print solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("Hours worked by Bill: ", x0.varValue)
    print("Hours worked by Jean: ", x1.varValue)
    print("Hours worked by John: ", x2.varValue)
    print("Hours worked by Mary: ", x3.varValue)
else:
    print("No optimal solution found")
```