how do we know that there is a subtraction of two close numbers thus making loss of significance?
there is no close numbers there is variable X
it could give use a close result or otherwise
and why multiplying and dividing by
makes it go away?
The point of the exercise is that in this form the value can be computed without the catastrophic loss of significant figures that occurs in the original computation when x is small due to the subtraction of two almost-equal numbers.
Or, to put it another way, No Subtraction!