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We explore the utilities of explicit negative examples in training neural language models. Negative examples here are incorrect words in a sentence, such as "barks" in "*The dogs barks". Neural language models are commonly trained only on positive examples, a set of sentences in the training data, but recent studies suggest that the models trained in this way are not capable of robustly handling complex syntactic constructions, such as long-distance agreement. In this paper, using English data, we first demonstrate that appropriately using negative examples about particular constructions (e.g., subject-verb agreement) will boost the model's robustness on them, with a negligible loss of perplexity. The key to our success is an additional margin loss between the log-likelihoods of a correct word and an incorrect word. We then provide a detailed analysis of the trained models. One of our findings is the difficulty of object-relative clauses for RNNs. We find that even with our direct learning signals the models still suffer from resolving agreement across an object-relative clause. Augmentation of training sentences involving the constructions somewhat helps, but the accuracy still does not reach the level of subject-relative clauses. Although not directly cognitively appealing, our method can be a tool to analyze the true architectural limitation of neural models on challenging linguistic constructions.