## Step 1: Define the variables and their attributes
We have four variables: 'hours worked by Hank', 'hours worked by Bill', 'hours worked by Jean', and 'hours worked by Dale'. Let's denote them as $H_H$, $H_B$, $H_J$, and $H_D$ respectively. Each variable has attributes related to dollar cost per hour and paperwork competence rating.

## 2: Formulate the objective function
The objective function to minimize is:
\[1.77H_H^2 + 8.58H_HH_B + 2.9H_HH_J + 9.51H_HH_D + 5.01H_B^2 + 3.71H_HH_J + 3.57H_BH_D + 3.62H_JH_D + 4.16H_D^2 + 7.54H_H + 6.0H_B + 7.17H_J + 8.16H_D\]

## 3: Define the constraints
1. $2H_H \leq 133$
2. $6H_H \leq 104$
3. $5H_B \leq 133$
4. $8H_B \leq 104$
5. $1H_J \leq 133$
6. $1H_J \leq 104$
7. $3H_D \leq 133$
8. $2H_D \leq 104$
9. $5H_B + 3H_D \geq 28$
10. $5H_B + 1H_J \geq 24$
11. $1H_J^2 + 3H_D^2 \geq 13$
12. $2H_H^2 + 5H_B^2 \geq 23$
13. $2H_H^2 + 1H_J^2 + 3H_D^2 \geq 19$
14. $2H_H^2 + 5H_B^2 + 3H_D^2 \geq 19$
15. $2H_H^2 + 1H_J^2 + 3H_D^2 \geq 28$
16. $2H_H + 5H_B + 3H_D \geq 28$
17. $2H_H + 5H_B + 1H_J + 3H_D \geq 28$
18. $8H_B + 1H_J \geq 26$
19. $6H_H + 8H_B + 1H_J + 2H_D \geq 26$
20. $-7H_B^2 + 1H_D^2 \geq 0$
21. $5H_B + 1H_J \leq 118$
22. $5H_B + 3H_D \leq 69$
23. $8H_B^2 + 2H_D^2 \leq 62$
24. $6H_H^2 + 2H_D^2 \leq 98$
25. $6H_H + 8H_B + 2H_D \leq 91$
26. $H_H$ is an integer
27. $H_B$ is an integer
28. $H_J$ is continuous
29. $H_D$ is continuous

## 4: Implement the problem in Gurobi
```python
import gurobi as gp

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

# Define the variables
H_H = m.addVar(name="H_H", vtype=gp.GRB.INTEGER)
H_B = m.addVar(name="H_B", vtype=gp.GRB.INTEGER)
H_J = m.addVar(name="H_J")
H_D = m.addVar(name="H_D")

# Define the objective function
m.setObjective(1.77*H_H**2 + 8.58*H_H*H_B + 2.9*H_H*H_J + 9.51*H_H*H_D + 
               5.01*H_B**2 + 3.71*H_B*H_J + 3.57*H_B*H_D + 3.62*H_J*H_D + 
               4.16*H_D**2 + 7.54*H_H + 6.0*H_B + 7.17*H_J + 8.16*H_D, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(2 * H_H <= 133)
m.addConstr(6 * H_H <= 104)
m.addConstr(5 * H_B <= 133)
m.addConstr(8 * H_B <= 104)
m.addConstr(1 * H_J <= 133)
m.addConstr(1 * H_J <= 104)
m.addConstr(3 * H_D <= 133)
m.addConstr(2 * H_D <= 104)
m.addConstr(5 * H_B + 3 * H_D >= 28)
m.addConstr(5 * H_B + 1 * H_J >= 24)
m.addConstr(1 * H_J**2 + 3 * H_D**2 >= 13)
m.addConstr(2 * H_H**2 + 5 * H_B**2 >= 23)
m.addConstr(2 * H_H**2 + 1 * H_J**2 + 3 * H_D**2 >= 19)
m.addConstr(2 * H_H**2 + 5 * H_B**2 + 3 * H_D**2 >= 19)
m.addConstr(2 * H_H**2 + 1 * H_J**2 + 3 * H_D**2 >= 28)
m.addConstr(2 * H_H + 5 * H_B + 3 * H_D >= 28)
m.addConstr(2 * H_H + 5 * H_B + 1 * H_J + 3 * H_D >= 28)
m.addConstr(8 * H_B + 1 * H_J >= 26)
m.addConstr(6 * H_H + 8 * H_B + 1 * H_J + 2 * H_D >= 26)
m.addConstr(-7 * H_B**2 + 1 * H_D**2 >= 0)
m.addConstr(5 * H_B + 1 * H_J <= 118)
m.addConstr(5 * H_B + 3 * H_D <= 69)
m.addConstr(8 * H_B**2 + 2 * H_D**2 <= 62)
m.addConstr(6 * H_H**2 + 2 * H_D**2 <= 98)
m.addConstr(6 * H_H + 8 * H_B + 2 * H_D <= 91)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("H_H: ", H_H.varValue)
    print("H_B: ", H_B.varValue)
    print("H_J: ", H_J.varValue)
    print("H_D: ", H_D.varValue)
else:
    print("The model is infeasible")
```