How many real numbers x satisfy the inequality below?

Let y = x^4 - 4x^2. Then the inequality becomes

|y - 6| > |y + 14|

|y - 6| > |y - (-14)|

Since |a-b| is the distance on the number line between a and b, this inequality just says, in words, "the distance between y and 6 is greater than or equal to the distance between y and -14", or in other words, "y is closer to -14 than it is to 6". So y is at or less than the midpoint between 6 and -14, and y must be less than or equal to -4. Since y = x^4 - 4x^2, we just need to determine when:

x^4 - 4x^2 < -4
x^4 - 4x^2 + 4 < 0
(x^2 - 2)^2 < 0

and since the left side above is a square, it cannot be negative, so we only have solutions when (x^2 - 2)^2 = 0, so when x^2 - 2 = 0, and x^2 = 2. That equation has two solutions, x = √2 and x = -√2, so the answer is C.