Typically, automatic Question Answering (QA) approaches use the question in its entirety in the search for potential answers. We argue that decomposing complex factoid questions into separate facts about their answers is beneficial to QA, since an answer candidate with support coming from multiple independent facts is more likely to be the correct one. We broadly categorize decomposable questions as parallel or nested, and we present a novel question decomposition framework for enhancing the ability of single-shot QA systems to answer complex factoid questions. Essential to the framework are components for decomposition recognition, question rewriting, and candidate answer synthesis and re-ranking. We discuss the interplay among these, with particular emphasis on decomposition recognition, a process which, we argue, can be sufficiently informed by lexico-syntactic features alone. We validate our approach to decomposition by implementing the framework on top of IBM Watson™, a state-of-the-art QA system, and showing a statistically significant improvement over its accuracy.