Would you consider using another's analysis as a basis for your own analysis on a poem/story plagiarism?
It might be plagiarism or not, depending on how you present it. If you do it in such a way that the ideas seem to the reader to be yours alone, then it is clearly plagiarism. It isn't the "word-for-word" issue at all. That is irrelevant. 

But if you quote the original work and then take off from there, extending the ideas of the others(s) and you provide a citation for it then it is clearly not plagiarism. Plagiarism is taking the ideas of another as if they were your own without attribution. It is always fine to extend the ideas of others, provided that it is clear that it is an extension and clear that you don't attribute the underlying ideas to yourself. 

Don't confuse this with copyright, which is more to do with the actual words used. 

However, if this is for a class, then the standards set by the professor may go beyond mere avoidance of plagiarism. So, while you may avoid plagiarism by using proper citation it isn't necessarily true that you meet the course standards. The judgement on that will only come from the instructor.