2007 AMC 10A Problem 20

Squaring $a + a^{-1} = 4$ gives $a^2 + 2 + a^{-2} = 16,$ so $a^2 + a^{-2} = 14.$ Squaring again gives $a^4 + 2 + a^{-4} = 196,$ so $a^4 + a^{-4} = 194.$ Thus, the correct answer is D.