1976 AHSME Problem 30

How many distinct ordered triples (x,y,z) satisfy the following equations? \begin{aligned} x + 2y + 4z &= 12 \\ xy + 4yz + 2xz &= 22 \\ xyz &= 6 \end{aligned} \textbf{(A) }\text{none}\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }4\qquad \textbf{(E) }6

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