Solution:

First, note that $$x^4-51x^2+50$$$$= (x^2-50)(x^2-1).$$

If $(x^2-50)(x^2-1) < 0$ means that one of the terms is negative.

Since $$x^2-50 < x^2-1,$$ it must be that x^2-50 < 0, x^2-1 >0. This means $1 < x^2 < 50,$ making $$1< |x| \leq 7,$$ resulting in $12$ solutions.

Thus, the correct answer is C .