According to the Remainder theorem, the fact that (x-2) is the factor of the polynomial MEANS that

the number x= 2 is the root of the polynomial.


So, we substitute  x= 2 into the polynomial and equate it to zero ( ! the root (!) ).


We get then this equation

    k*2^3 - k*2^2 + 2 + 2 = 0.


It is the equation to find the value of "k".  

Simplify it

    8k - 4k + 2 + 2 = 0

    4k      + 4     = 0

    4k              = -4

     k              = -4/4 = -1.


ANSWER.  k = -1.