## Step 1: Define the optimization problem
The problem requires minimizing the objective function: $1 \times (\text{hours worked by Bill})^2 + 9 \times (\text{hours worked by Bill}) \times (\text{hours worked by Peggy}) + 8 \times (\text{hours worked by Peggy})^2 + 1 \times (\text{hours worked by Peggy}) \times (\text{hours worked by Hank}) + 7 \times (\text{hours worked by Peggy}) + 7 \times (\text{hours worked by Hank})$.

## 2: Identify the variables
Let $x_B$ be the hours worked by Bill, $x_P$ be the hours worked by Peggy, and $x_H$ be the hours worked by Hank.

## 3: Formulate the objective function
The objective function to minimize is: $x_B^2 + 9x_Bx_P + 8x_P^2 + x_Px_H + 7x_P + 7x_H$.

## 4: Define the constraints
The constraints are:
- $x_B \geq 0$, $x_P \geq 0$, $x_H \geq 0$ (non-negativity constraints)
- $0.46x_B \leq 144$
- $1.82x_B \leq 116$
- $6.95x_B \leq 152$
- $8.81x_B \leq 76$
- $4.17x_B \leq 99$
- $6.72x_P \leq 144$
- $2.03x_P \leq 116$
- $1.29x_P \leq 152$
- $1.9x_P \leq 76$
- $10.11x_P \leq 99$
- $10.12x_H \leq 144$
- $5.95x_H \leq 116$
- $12.71x_H \leq 152$
- $9.2x_H \leq 76$
- $0.97x_H \leq 99$
- $0.46x_B^2 + 10.12x_H^2 \geq 39$
- $6.72x_P + 10.12x_H \geq 18$
- $0.46x_B + 6.72x_P + 10.12x_H \geq 18$
- $1.82x_B + 2.03x_P \geq 16$
- $2.03x_P + 5.95x_H \geq 36$
- $1.82x_B + 2.03x_P + 5.95x_H \geq 34$
- $6.95x_B + 12.71x_H \geq 31$
- $6.95x_B + 1.29x_P \geq 47$
- $6.95x_B + 1.29x_P + 12.71x_H \geq 47$
- $8.81x_B^2 + 9.2x_H^2 \geq 11$
- $8.81x_B + 1.9x_P \geq 14$
- $8.81x_B + 1.9x_P + 9.2x_H \geq 14$
- $4.17x_B + 10.11x_P \geq 31$
- $10.11x_P^2 + 0.97x_H^2 \geq 30$
- $4.17x_B + 10.11x_P + 0.97x_H \geq 30$
- $5x_P - 5x_H \geq 0$
- $2.03x_P + 5.95x_H \leq 88$
- $1.82x_B^2 + 2.03x_P^2 \leq 54$
- $4.17x_B + 10.11x_P \leq 62$
- $4.17^2x_B^2 + 10.11^2x_P^2 + 0.97^2x_H^2 \leq 96$

## 5: Implement the problem in Gurobi
```python
import gurobi

def optimize_problem():
    model = gurobi.Model()

    # Define variables
    x_B = model.addVar(lb=0, name="hours_worked_by_Bill")
    x_P = model.addVar(lb=0, name="hours_worked_by_Peggy")
    x_H = model.addVar(lb=0, name="hours_worked_by_Hank")

    # Objective function
    model.setObjective(x_B**2 + 9*x_B*x_P + 8*x_P**2 + x_P*x_H + 7*x_P + 7*x_H)

    # Constraints
    model.addConstr(0.46 * x_B <= 144)
    model.addConstr(1.82 * x_B <= 116)
    model.addConstr(6.95 * x_B <= 152)
    model.addConstr(8.81 * x_B <= 76)
    model.addConstr(4.17 * x_B <= 99)

    model.addConstr(6.72 * x_P <= 144)
    model.addConstr(2.03 * x_P <= 116)
    model.addConstr(1.29 * x_P <= 152)
    model.addConstr(1.9 * x_P <= 76)
    model.addConstr(10.11 * x_P <= 99)

    model.addConstr(10.12 * x_H <= 144)
    model.addConstr(5.95 * x_H <= 116)
    model.addConstr(12.71 * x_H <= 152)
    model.addConstr(9.2 * x_H <= 76)
    model.addConstr(0.97 * x_H <= 99)

    model.addConstr(0.46 * x_B**2 + 10.12 * x_H**2 >= 39)
    model.addConstr(6.72 * x_P + 10.12 * x_H >= 18)
    model.addConstr(0.46 * x_B + 6.72 * x_P + 10.12 * x_H >= 18)
    model.addConstr(1.82 * x_B + 2.03 * x_P >= 16)
    model.addConstr(2.03 * x_P + 5.95 * x_H >= 36)
    model.addConstr(1.82 * x_B + 2.03 * x_P + 5.95 * x_H >= 34)
    model.addConstr(6.95 * x_B + 12.71 * x_H >= 31)
    model.addConstr(6.95 * x_B + 1.29 * x_P >= 47)
    model.addConstr(6.95 * x_B + 1.29 * x_P + 12.71 * x_H >= 47)
    model.addConstr(8.81 * x_B**2 + 9.2 * x_H**2 >= 11)
    model.addConstr(8.81 * x_B + 1.9 * x_P >= 14)
    model.addConstr(8.81 * x_B + 1.9 * x_P + 9.2 * x_H >= 14)
    model.addConstr(4.17 * x_B + 10.11 * x_P >= 31)
    model.addConstr(10.11 * x_P**2 + 0.97 * x_H**2 >= 30)
    model.addConstr(4.17 * x_B + 10.11 * x_P + 0.97 * x_H >= 30)
    model.addConstr(5 * x_P - 5 * x_H >= 0)
    model.addConstr(2.03 * x_P + 5.95 * x_H <= 88)
    model.addConstr(1.82 * x_B**2 + 2.03 * x_P**2 <= 54)
    model.addConstr(4.17 * x_B + 10.11 * x_P <= 62)
    model.addConstr(4.17**2 * x_B**2 + 10.11**2 * x_P**2 + 0.97**2 * x_H**2 <= 96)

    # Solve the problem
    model.optimize()

    # Output the solution
    if model.status == gurobi.GRB_OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Bill: {x_B.varValue}")
        print(f"Hours worked by Peggy: {x_P.varValue}")
        print(f"Hours worked by Hank: {x_H.varValue}")
        print(f"Objective function value: {model.objVal}")
    elif model.status == gurobi.GRB_INFEASIBLE:
        print("No feasible solution exists.")
    else:
        print("No optimal solution exists.")

optimize_problem()
```