One of the main problems in undergraduate research in pure mathematics is that of determining a problem that is, at once, interesting to and capable of solution by a student who has completed only the calculus sequence. It is also desirable that the problem should present something new, since novelty and originality greatly increase the enthusiasm of the student if no one has previously solved it. One of the most accessible topics for undergraduate research is the theory of polynomials. There are a great number of good textbooks on the theory of equations that can be used for preliminary reading and which will prepare the student for work with more advanced scientific texts. At the same time, the student is familiar with polynomials in terms of their graphs and basic properties, as discussed in calculus. This intertwining of familiar material and new algebraic approach helps to create an atmosphere of comfort and interest for the student. In addition, the use of computer software incorporates the flavor of modern science and eliminates long calculations. This paper focuses on the stages of development of a research project on polynomials and the results that were obtained. The problem of characterizing cubic polynomials in terms of their coefficients arose during the study of the graphs of polynomials in pre-calculus. The solution to this problem was published, and that work served as a starting point for the following problem, which was proposed to the student: "To describe, in terms of the coefficients, all possible shapes of polynomials of degree higher than three." In this article, the authors present a complete description of the shapes of fourth-degree polynomials in terms of their coefficients. (Contains 6 figures.)