## To solve this problem, we can use a combination of algebra and logical reasoning. Let's break it down step by step:

1. We know that Pauline spends 4 hours selling a used car and 6 hours selling a new car. So, in order to maximize her bonus, she should work only 4 hours on a used car and the remaining time on selling new cars.

2. The total number of hours she can work in a week is limited to 40 hours.

3. Let's assume she sells x used cars and y new cars.

4. From step 1, we know that she spends 4 hours on each used car, so the total time spent on selling used cars will be 4x hours.

5. The time spent on selling new cars will be 6y hours.

6. From steps 4 and 5, we can set up the equation: 4x + 6y = 40.

7. We also know that in order to receive a bonus, she must sell at least one used car and four new cars each week.

8. So, we can set up another equation based on the requirement: x ≥ 1 and y ≥ 4.

9. Now, we need to determine the values of x and y that maximize her bonus.

10. In this case, the bonus for selling a used car is $180.00 and the bonus for selling a new car is $290.00.

11. To maximize her bonus, she should aim to maximize the total amount earned for selling new cars.

12. As per the given data, she earns $180.00 selling each used car and $290.00 selling each new car.

13. We can calculate the amount earned for selling used cars: 180x.

14. We can calculate the amount earned for selling new cars: 290y.

15. The total bonus earned can be represented as: Total Bonus = 180x + 290y.

16. To maximize the total bonus earned, we need to maximize the value of Total Bonus.

17. Therefore, our objective is to maximize the function f(x, y) = 180x + 290y.

18. Now, we have two equations:

- 4x + 6y = 40 (from step 6)

- x ≥ 1 and y ≥ 4 (from step 8)

19. To find the values of x and y that maximize the function f(x, y), we can solve the system of equations formed by steps 6 and 8 using either substitution or elimination method.

20. Once we find the values of x and y, we can calculate the corresponding bonus using the formulas mentioned in step 10.

In conclusion, to maximize her bonus, Pauline should aim to sell one used car and six new cars, which will result in a bonus of $180.00 for the used car and $290.00 for each new car.