You won't come across the term "Marxist analysis" very often unless you spend a lot of your time reading radical Leftist literature, but to celebrate Independence Day Jacobin Magazine published a Marxist analysis of the US constitution written in 1990 by Bertell Ollman, who teaches Marxist theory at New York University.
When Marxism is mentioned its usually in terms of Socialism, even though that's just one aspect of Marxism. There are also the theories Marx and Engels laid out in their many writings that established new ways of analyzing many things -- art, music, literature, society, and in this case a historical document.
As you read through it and see what kinds of things it takes into account, how it weighs different aspects of its subject, the assumptions and values it makes use of, you'll notice that although it's pretty "radical" sounding, a lot of it isn't that far different from a lot of what you'd read in mainstream academia. That's because mainstream academia has been heavily influenced by Marxism -- Marxist theory and Marxist academics. It's just not widely recognized as such.