What is the least common denominator of 4/x * -4/2x^2?

2x^2

Justification:

First we must make the expressions clear.

It seems you are asking about the least common denominator of 4/x and - 4/ (2x^2).

I will not simpligy the second denominator and will find the least common denominator of x and 2x^2.

The least common factor is the simplest (smallest) factor that contain all the factors given.

So, the least common denominator of both x and 2x^2 is the simplest monomial that contains them.

To find that least common factor you take the  product of:

- least common multiple of the coefficients: in this case the coefficients are 1 and 2, so the least common coefficient is 2.

- the product of the different letter raised to the highest power: in this case the only letter is x and the highest power is 2, so it is x^2

Therefore the lease common denominator is the product of 2 and x^2 which is 2x^2.