The first is from Intel, and it asks you to take the "Intel Computer Personality Quiz." You'll do by clicking on the Sponsored Link at the bottom right corner of the FarmVille game page which will load a new window, where you'll answer a series of personality questions, about computers of course, until you are told that you earned your free Farm Cash in the top right corner of this new window.
The second new promotion is from McDonald's. Following the same procedure, this promotion, when clicked, loads a new window where you'll answer a quick poll, and will receive your Farm Cash in the same way as above - just keep an eye out for the phrase stating that you've earned your Farm Cash, and you can close the window at any time thereafter.
Finally, Bing is at it again, offering another promotion giving users 2 Farm Cash for participating. This promotion is a bit more complex, as it asks you to play the Word Challenge at Club Bing in order to earn your free Farm Cash. Please note that this promotion is different from the ones we saw released in July and November of this year.
As these Farm Cash promotions tend to appear and then disappear at random (and rather quickly at that), it's possible (nay, likely) that you won't see these particular promotions the next time you log into the game. However, with enough refreshes, or perhaps a bit of luck, they are sure to cycle around to your farm in time, allowing you cash in on some free Farm Cash.
Regardless of the promotion you click on, remember that your Farm Cash will automatically be added to your in-game total, it will NOT go to your Gift Box. Simply refresh the page after each promotion and you will see your updated (read: larger) Farm Cash total.
Let us know what you think of these cross-promotions in the comments. Which one is your favorite? Do you think they take too long to complete, or that the reward should be higher?