Can I use a business or product name and copyright protect it if part of the name is already included in a copyright protected business?

Business and product names cannot be protected under the law of copyright. Such names are covered by the law of trademark. As a general matter, assuming another person (the "senior user") owns a trademark in his business or product name, you (the "junior user") cannot adopt the same or a colorably similar name if the result will be that an appreciable number of ordinary consumers will be confused into believing, incorrectly, that the junior user's products are authorized, sponsored by or originate with the senior user. Whether or not such confusion will occur is an intensely fact-based determination. Accordingly, you will need to confer with an attorney to determine the appropriateness of your proposed mark. If the senior user's mark is "strong" or "famous," it will generally be inadvisable to adopt a junior mark that is close to it. Your attorney can properly advise you regarding this matter.