## Step 1: Define the variables and the objective function
The variables are 'hours worked by Hank', 'hours worked by Laura', 'hours worked by Mary', and 'hours worked by Peggy'. Let's denote them as $H$, $L$, $M$, and $P$ respectively. The objective function to minimize is $9.66H^2 + 2.25HL + 5.78HM + 9.13HP + 7.01L^2 + 4.14M^2 + 3.29P^2 + 8.94M$.

## 2: Define the constraints
The constraints given are:
- $12H \leq 184$ (Hank's paperwork competence rating)
- $16H \leq 229$ (Hank's likelihood to quit index)
- $11L \leq 184$ (Laura's paperwork competence rating)
- $19L \leq 229$ (Laura's likelihood to quit index)
- $7M \leq 184$ (Mary's paperwork competence rating)
- $14M \leq 229$ (Mary's likelihood to quit index)
- $4P \leq 184$ (Peggy's paperwork competence rating)
- $13P \leq 229$ (Peggy's likelihood to quit index)
- $12H + 7M \geq 32$ (combined paperwork competence rating from Hank and Mary)
- $11L + 7M \geq 17$ (combined paperwork competence rating from Laura and Mary)
- $11L + 4P \geq 21$ (combined paperwork competence rating from Laura and Peggy)
- $12H + 11L + 7M \geq 40$ (combined paperwork competence rating from Hank, Laura, and Mary)
- $12H + 11L + 7M + 4P \geq 40$ (combined paperwork competence rating from all)
- $16H + 14M \geq 47$ (combined likelihood to quit index from Hank and Mary)
- $14M^2 + 13P^2 \geq 56$ (combined likelihood to quit index from Mary and Peggy)
- $19L^2 + 14M^2 \geq 22$ (combined likelihood to quit index from Laura and Mary)
- $19L + 13P \geq 32$ (combined likelihood to quit index from Laura and Peggy)
- $16H + 19L \geq 36$ (combined likelihood to quit index from Hank and Laura)
- $16H + 13P \geq 20$ (combined likelihood to quit index from Hank and Peggy)
- $16H + 19L + 14M + 13P \geq 20$ (combined likelihood to quit index from all)
- $-2M + 6P \geq 0$ (relationship between Mary and Peggy's hours)
- $-8L^2 + 7M^2 \geq 0$ (relationship between Laura and Mary's hours)
- $10H^2 - 8P^2 \geq 0$ (relationship between Hank and Peggy's hours)
- $11L^2 + 7M^2 \leq 184$ (paperwork competence rating limit)
- $12H + 4P \leq 103$ (paperwork competence rating limit from Hank and Peggy)
- $12H + 11L \leq 52$ (paperwork competence rating limit from Hank and Laura)
- $11L + 4P \leq 146$ (paperwork competence rating limit from Laura and Peggy)
- $12H^2 + 11L^2 + 4P^2 \leq 125$ (paperwork competence rating limit from squared hours)

## 3: Specify variable types
- $H$ can be fractional
- $L$ must be an integer
- $M$ must be an integer
- $P$ can be fractional

## 4: Implement in Gurobi
We will use Gurobi's Python API to model and solve this problem.

```python
import gurobi as gp

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

# Define variables
H = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, name="H")  # Fractional
L = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, name="L", vtype=gp.GRB.INTEGER)  # Integer
M = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, name="M", vtype=gp.GRB.INTEGER)  # Integer
P = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, name="P")  # Fractional

# Objective function
m.setObjective(9.66 * H**2 + 2.25 * H * L + 5.78 * H * M + 9.13 * H * P + 
               7.01 * L**2 + 4.14 * M**2 + 3.29 * P**2 + 8.94 * M, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(12 * H <= 184, name="H_paperwork")
m.addConstr(16 * H <= 229, name="H_quit")
m.addConstr(11 * L <= 184, name="L_paperwork")
m.addConstr(19 * L <= 229, name="L_quit")
m.addConstr(7 * M <= 184, name="M_paperwork")
m.addConstr(14 * M <= 229, name="M_quit")
m.addConstr(4 * P <= 184, name="P_paperwork")
m.addConstr(13 * P <= 229, name="P_quit")

m.addConstr(12 * H + 7 * M >= 32, name="H_M_paperwork")
m.addConstr(11 * L + 7 * M >= 17, name="L_M_paperwork")
m.addConstr(11 * L + 4 * P >= 21, name="L_P_paperwork")
m.addConstr(12 * H + 11 * L + 7 * M >= 40, name="H_L_M_paperwork")
m.addConstr(12 * H + 11 * L + 7 * M + 4 * P >= 40, name="all_paperwork")

m.addConstr(16 * H + 14 * M >= 47, name="H_M_quit")
m.addConstr(14 * M**2 + 13 * P**2 >= 56, name="M_P_quit")
m.addConstr(19 * L**2 + 14 * M**2 >= 22, name="L_M_quit")
m.addConstr(19 * L + 13 * P >= 32, name="L_P_quit")
m.addConstr(16 * H + 19 * L >= 36, name="H_L_quit")
m.addConstr(16 * H + 13 * P >= 20, name="H_P_quit")
m.addConstr(16 * H + 19 * L + 14 * M + 13 * P >= 20, name="all_quit")

m.addConstr(-2 * M + 6 * P >= 0, name="M_P_relationship")
m.addConstr(-8 * L**2 + 7 * M**2 >= 0, name="L_M_relationship")
m.addConstr(10 * H**2 - 8 * P**2 >= 0, name="H_P_relationship")

m.addConstr(11 * L**2 + 7 * M**2 <= 184, name="paperwork_limit")
m.addConstr(12 * H + 4 * P <= 103, name="H_P_limit")
m.addConstr(12 * H + 11 * L <= 52, name="H_L_limit")
m.addConstr(11 * L + 4 * P <= 146, name="L_P_limit")
m.addConstr(12 * H**2 + 11 * L**2 + 4 * P**2 <= 125, name="paperwork_limit_squared")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("H: ", H.varValue)
    print("L: ", L.varValue)
    print("M: ", M.varValue)
    print("P: ", P.varValue)
else:
    print("No optimal solution found")
```