The phrase "with a good background" can be an important descriptor in various situations, such as job applications or academic achievements. However, sometimes we might want to vary our vocabulary and expand our options. Some synonyms for "with a good background" could be "having a solid foundation," "possessing significant experience," "equipped with a strong history," "well-versed," or "knowledgeable in." All of these phrases convey a depth of understanding and capability that is desirable in many contexts. It's always good to have a range of phrases to choose from when attempting to communicate our qualifications and strengths.