In what ways is the tragic outcome of the novel caused by carelessness? And what quotes could be used to back this up?

You can consider a few things. The fact that Tom tells George that Gatsby ran Myrtle over begins the final connection to the killing of Gatsby. Wilson takes Tom's word at face value. He doesn't question Gatsby before he kills him. Wilson's world has fallen apart and I don't even think that he cares much for himself.