1983 AHSME Problem 18

Let f be a polynomial function such that, for all real x, f(x^2 + 1) = x^4 + 5x^2 + 3. For all real x, f(x^2-1) is

\textbf{(A)}\ x^4+5x^2+1\qquad \textbf{(B)}\ x^4+x^2-3\qquad \textbf{(C)}\ x^4-5x^2+1\qquad \textbf{(D)}\ x^4+x^2+3\qquad \textbf{(E)}\ \text{none of these}

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