1965 AHSME Problem 40

Let n be the number of integer values of x such that P = x^4 + 6x^3 + 11x^2 + 3x + 31 is the square of an integer. Then n is:

\textbf{(A)}\ 4 \qquad \textbf{(B) }\ 3 \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\ 1 \qquad \textbf{(E) }\ 0

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.