## Step 1: Define the optimization problem and the objective function
The goal is to minimize the objective function: $4 \cdot (hours\_worked\_by\_Laura)^2 + 8 \cdot hours\_worked\_by\_Laura \cdot hours\_worked\_by\_George + 2 \cdot hours\_worked\_by\_George \cdot hours\_worked\_by\_Paul + 2 \cdot (hours\_worked\_by\_Paul)^2 + 4 \cdot hours\_worked\_by\_Paul$.

## Step 2: Define the variables
Let $L$ be the hours worked by Laura, $G$ be the hours worked by George, and $P$ be the hours worked by Paul.

## Step 3: Define the constraints
1. $7L \leq 76$
2. $1L \leq 82$
3. $9L \leq 130$
4. $3G \leq 76$
5. $10G \leq 82$
6. $7G \leq 130$
7. $3P \leq 76$
8. $9P \leq 82$
9. $1P \leq 130$
10. $7L^2 + 3G^2 \geq 24$
11. $7L + 3P \geq 23$
12. $7L^2 + 3G^2 + 3P^2 \geq 23$
13. $7L + 3G + 3P \geq 23$
14. $1L^2 + 10G^2 \geq 19$
15. $10G + 9P \geq 26$
16. $1L + 10G + 9P \geq 26$
17. $7G + 1P \geq 22$
18. $9L^2 + 7G^2 \geq 38$
19. $9L + 7G + 1P \geq 34$
20. $9L + 7G + 1P \geq 34$
21. $-7L^2 + 8G^2 \geq 0$
22. $7L + 3G \leq 39$
23. $7L^2 + 3G^2 + 3P^2 \leq 64$
24. $10G + 9P \leq 43$
25. $1L + 10G + 9P \leq 58$

## Step 4: Implement the optimization problem using Gurobi
```python
import gurobi

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

# Define the variables
L = m.addVar(lb=-gurobi.GRB.INFINITY, name="hours_worked_by_Laura")
G = m.addVar(lb=-gurobi.GRB.INFINITY, name="hours_worked_by_George")
P = m.addVar(lb=-gurobi.GRB.INFINITY, name="hours_worked_by_Paul")

# Define the objective function
m.setObjective(4*L**2 + 8*L*G + 2*G*P + 2*P**2 + 4*P, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(7*L <= 76)
m.addConstr(1*L <= 82)
m.addConstr(9*L <= 130)
m.addConstr(3*G <= 76)
m.addConstr(10*G <= 82)
m.addConstr(7*G <= 130)
m.addConstr(3*P <= 76)
m.addConstr(9*P <= 82)
m.addConstr(1*P <= 130)
m.addConstr(7*L**2 + 3*G**2 >= 24)
m.addConstr(7*L + 3*P >= 23)
m.addConstr(7*L**2 + 3*G**2 + 3*P**2 >= 23)
m.addConstr(7*L + 3*G + 3*P >= 23)
m.addConstr(1*L**2 + 10*G**2 >= 19)
m.addConstr(10*G + 9*P >= 26)
m.addConstr(1*L + 10*G + 9*P >= 26)
m.addConstr(7*G + 1*P >= 22)
m.addConstr(9*L**2 + 7*G**2 >= 38)
m.addConstr(9*L + 7*G + 1*P >= 34)
m.addConstr(9*L + 7*G + 1*P >= 34)
m.addConstr(-7*L**2 + 8*G**2 >= 0)
m.addConstr(7*L + 3*G <= 39)
m.addConstr(7*L**2 + 3*G**2 + 3*P**2 <= 64)
m.addConstr(10*G + 9*P <= 43)
m.addConstr(1*L + 10*G + 9*P <= 58)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Laura: ", L.varValue)
    print("Hours worked by George: ", G.varValue)
    print("Hours worked by Paul: ", P.varValue)
else:
    print("The model is infeasible")
```