Suppose you do work on an object from rest to a certain speed v. why does it take four times the amount of work to increase its speed from rest to a speed of 2v?

it is due to friction.

Explanation: if you were to push an object that is at rest, there is static friction that you would first need to overcome. An example would be pushing a car as it is in neutral. Pushing it takes more force and work to overcome that friction.
But once it overcomes static, it will then turn to kinetic friction. Once the car starts to move, you notice that it is easier to push. This is due to the fact that kinetic friction would equal normal force times the constant while the static friction is greater than or equal to the normal force times the constant.

Hope this helps!!!