Fully agreed the answer should be Gollum/Smeagol or The Ring. It's pretty fundamental to the story that it's Gollums obsession with The Ring that kills him not one of the hobbits.
In the book, Shadowfax does not kick him in the fire. Denethor puts himself there, and he dies there....and a palantir falls out of his robe. There is no running off the end of the precipice. Your quiz should clarify that you mean IN THE MOVIE.
as mentioned by others, it should be specified that this quiz is referring to the movie, not the book. For instance, in the book Grima is killed by the hobbits after he killed Saruman, Denethor commits suicide with no assistance from Shadowfax. Also, Frodo does not kill Gollum in either the book or the movie; Gollum falls into Mount Doom after wresting the Ring from Frodo.
Yeah, and Gothmog is from the Simarilion, and is killed by the king of Gondolin, aka Madril with Glamdring and his life, much like Gandalf. (Gothmog was a Balrog). Also the Balrog Gandalf killed was known as Durin's Bane because it killed Durin II. P.S Denathor did kill himself in both the book and the movie.