Below are the graphs of y = x^4 + 3x^2 - x + 1 and y = 3. as you can clearly see, they intersect only twice. namely at x = -0.6337532429 and x = 0.8738358889. thus there is something wrong with the question.
as you can see, the graph looks like a parabola, so, in fact, f(x) = c at most twice for any c
the question does not have a "prove or disprove" instruction, so i would not really recommend it. just tell your professor that there is something wrong with the question.
you can use IVT to prove there is at least 2. to disprove that there are at least 3 cannot be done with IVT as far as i can see, but we can disprove it (if that's allowed) using Rolle's theorem