DAVENPORT, Iowa, Jan. 28 — Senator Hillary Rodham Clinton was in a Cedar Rapids living room crammed with Iowa voters on Saturday night when a voice from the back asked why the Clintons were such polarizing figures in American politics. The room grew still, except for the crackle of the fireplace. “I could give you a flip answer that, you know, some people are going to like me, and some people are not,” Mrs. Clinton said, smiling and sipping from a cup of water. “But I think it’s bigger and deeper than that.” For the next few minutes, Mrs. Clinton...