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Financial engineering is primarily the application of mathematics and to a certain degree programming skills to develop financial models. These models in turn help you use existing financial data and instruments and find ways of leveraging them to better exploit the financial markets.
Although the name is misleading, financial engineering has nothing to do with the traditional fields of engineering, although certain skills like applied mathematics might be transferrable. Additionally as far as I know, financial engineering is graduate level program.
If I were you, I would try to sample a few finance, applied math/stats and programming courses to see if I liked any of them, before I decided to pursue a degree in financial engineering. It also depends on why/what do you envision yourself doing in finance going ahead.
Hope that helps answer some of your questions.