## Step 1: Define the variables and the objective function
The variables are 'hours worked by Ringo', 'hours worked by Hank', 'hours worked by Peggy', and 'hours worked by Laura'. Let's denote them as $R$, $H$, $P$, and $L$ respectively. The objective function to minimize is $2.41R^2 + 5.49RH + 1.74RP + 6.27RL + 9.86H^2 + 1.8HL + 4.41P^2 + 1.09PL + 3.96L^2 + 8.0R + 9.94H + 9.96P + 4.91L$.

## Step 2: Define the constraints
1. $0.71R \leq 391$
2. $0.04R \leq 305$
3. $0.32H \leq 391$
4. $0.82H \leq 305$
5. $0.91P \leq 391$
6. $0.66P \leq 305$
7. $0.52L \leq 391$
8. $0.19L \leq 305$
9. $0.91P^2 + 0.52L^2 \geq 42$
10. $0.32H + 0.52L \geq 64$
11. $0.71R + 0.32H \geq 67$
12. $0.71R + 0.32H + 0.91P + 0.52L \geq 67$
13. $0.82H + 0.66P \geq 69$
14. $0.04R^2 + 0.66P^2 \geq 32$
15. $0.66P + 0.19L \geq 26$
16. $0.04R + 0.19L \geq 40$
17. $0.04R + 0.82H + 0.66P + 0.19L \geq 40$
18. $R - 6P \geq 0$
19. $6R - 10L \geq 0$
20. $9H^2 - 3P^2 + 10L^2 \geq 0$
21. $0.32H + 0.91P \leq 258$
22. $0.71R + 0.52L \leq 321$
23. $0.32H^2 + 0.52L^2 \leq 243$
24. $0.32H + 0.91P + 0.52L \leq 154$
25. $0.04R^2 + 0.19L^2 \leq 163$
26. $0.66P^2 + 0.19L^2 \leq 186$
27. $0.82H + 0.19L \leq 143$
28. $0.04R^2 + 0.82H^2 + 0.66P^2 \leq 245$

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

```python
import gurobi as gp

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

# Define the variables
R = m.addVar(lb=0, name="Ringo")
H = m.addVar(lb=0, name="Hank")
P = m.addVar(lb=0, name="Peggy")
L = m.addVar(lb=0, name="Laura")

# Define the objective function
m.setObjective(2.41*R**2 + 5.49*R*H + 1.74*R*P + 6.27*R*L + 
               9.86*H**2 + 1.8*H*L + 4.41*P**2 + 1.09*P*L + 
               3.96*L**2 + 8.0*R + 9.94*H + 9.96*P + 4.91*L, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(0.71*R <= 391)
m.addConstr(0.04*R <= 305)
m.addConstr(0.32*H <= 391)
m.addConstr(0.82*H <= 305)
m.addConstr(0.91*P <= 391)
m.addConstr(0.66*P <= 305)
m.addConstr(0.52*L <= 391)
m.addConstr(0.19*L <= 305)
m.addConstr(0.91*P**2 + 0.52*L**2 >= 42)
m.addConstr(0.32*H + 0.52*L >= 64)
m.addConstr(0.71*R + 0.32*H >= 67)
m.addConstr(0.71*R + 0.32*H + 0.91*P + 0.52*L >= 67)
m.addConstr(0.82*H + 0.66*P >= 69)
m.addConstr(0.04*R**2 + 0.66*P**2 >= 32)
m.addConstr(0.66*P + 0.19*L >= 26)
m.addConstr(0.04*R + 0.19*L >= 40)
m.addConstr(0.04*R + 0.82*H + 0.66*P + 0.19*L >= 40)
m.addConstr(R - 6*P >= 0)
m.addConstr(6*R - 10*L >= 0)
m.addConstr(9*H**2 - 3*P**2 + 10*L**2 >= 0)
m.addConstr(0.32*H + 0.91*P <= 258)
m.addConstr(0.71*R + 0.52*L <= 321)
m.addConstr(0.32*H**2 + 0.52*L**2 <= 243)
m.addConstr(0.32*H + 0.91*P + 0.52*L <= 154)
m.addConstr(0.04*R**2 + 0.19*L**2 <= 163)
m.addConstr(0.66*P**2 + 0.19*L**2 <= 186)
m.addConstr(0.82*H + 0.19*L <= 143)
m.addConstr(0.04*R**2 + 0.82*H**2 + 0.66*P**2 <= 245)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Ringo: ", R.varValue)
    print("Hank: ", H.varValue)
    print("Peggy: ", P.varValue)
    print("Laura: ", L.varValue)
else:
    print("The model is infeasible")
```