Here is the problem: you are given that the area of a rectangle is given by and that the length of one side is .
Since the area of a rectangle is given by the product of the two lengths, we find the length of the other side by dividing, and since we divide by a fraction by "inverting" and multiplying that is .
That you have right.
The perimeter of the rectangle is 2 times the sum of those two lengths. The sum of the lengths alone is . To add we get the common denominator: multiply numerator and denominator of the first fraction by x-1 and the second fraction by x+1:
Now multiply that by 2: the perimeter is , exactly what you have.
What numbers did you try? If we take x= 3, the side given is (3-2)(3+1)= 1/4 and the area is (18- 21+ 6)/(9-1)= 3/8 so the other side is (3/8)(4/1)= 3/2. Then the perimeter is 3/2+ 1/4+ 3/2+ 1/4= (6+ 1+ 6+ 1)/4= 14/4= 7/2, while the formula you got gives (54- 24- 2)/(9- 1)= 28/8= 7/2, exactly the same.