Question 366845
Dividing the polynomials gives : 
 

2x^3 - 5x^2 - kx + 3  | 2x - 1
------------------------------
2x^3 - x^2............| x^2 - 2x - (k+2)/2
----------------------|
-4x^2 - kx + 3........|
-4x^2 + 2x............|
----------------------|
-(k+2)x + 3...........|
-(k+2)x + (k+2)/2.....|
----------------------|
3 - (k+2)/2
 
 
the remainder should equals zero, hence k+2 = 6, k = 4
 
we can check another way by the fact the 1/2 has to be a solution : 
 
2/8 - 5/4 - 2 + 3 = 1/4 - 5/4 + 1 = -4/4 + 1 = 0
 
 
the remaining polynomial is : x^2 - 2x - 3 = (x-3)(x+1) by Viete's formula.
 
 
the 2 other roots are : 3 and -1
 
 
check : for x=3 : 2*27 - 5*9 - 4*3 + 3 = 54 - 45 - 12 + 3 = 9 - 12 + 3 = 0
 
x=-1 : -2 - 5 + 4 + 3 = -7 + 7 = 0