Extraneous solutions are valid solutions to the equation, but they make no sense for the reality of the question.
An example would be a quadratic equation with a positive and negative root when the variable in the equation represents a length, which cannot be negative.
Every polynomial equation has at least one root (Fundamental Theorem of Algebra). One or more of those roots may be complex so that if such an equation were modelling a real problem, the problem would have no real solution, Right off, I can't think of a good example. What I can also say is that there are some problems in electricity and magnetism for which complex roots do, in fact, make sense (although I am not enough of an engineer to explain this further).
