The comments were made on the latest episode, which aired last Sunday in the UK, in the show's news segment. Jeremy Clarkson, James May and Richard Hammond were talking about the all-new, and all-Mexican, sports car, the Mastretta MXT.
Richard Hammond started out the mocking by saying a Mexican car would be like Mexican people, saying their cars are just going to be:
"Lazy, feckless, flatulent, leaning against a fence asleep looking at a cactus with a blanket with a hole in the middle on as a coat."
James May then added more fuel to the flame by saying,
"They can't do food the Mexicans can they, because it's all like sick, with cheese on it."
Jeremy Clarkson then went on to say that the show isn't likely to get complaints from the Mexican Embassy, saying
"The Ambassador is just going to be sitting there with a remote control [sleeping], they won't complain."
Since then, Mexican Ambassador, Eduardo Medina-Mora Icaza, has filed a formal complaint against the show, demanding a public apology.
Mexican Ambassador, Eduardo Medina-Mora Icaza, was quoted as saying in the Daily Mail,
"These offensive, xenophobic and humiliating remarks only serve to reinforce negative stereotypes and perpetuate prejudice against Mexico and its people."
We found a clip of the incident in question floating around on YouTube if you want to see it.
The BBC has since received over 70 complaints about the matter, but says it will respond to the Mexican Embassy directly.