Start with this:
              
       ---------------------- 
2x - 1 )6x³ + 19x² +   x - 10
        

2x divides into 6x³ to give 3x²
Put that at the top, above the 19x²  

               3x²       
       ---------------------- 
2x - 1 )6x³ + 19x² +   x - 10
      
Multiply the 3x² by the 2x - 1 to get 6x³ - 3x²,
so write that at the bottom and draw a line:


               3x² 
       ---------------------- 
2x - 1 )6x³ + 19x² +   x - 10
        6x³ -  3x²
        ----------
           
Now subtract, by mentally changing the signs on
the bottom row.  We get 22x², so we write that
at the bottom, and bring down the next term x:

                       
               3x² 
       ---------------------- 
2x - 1 )6x³ + 19x² +   x - 10
        6x³ -  3x²
        ----------
              22x² +   x
              
2x divides into 22x² to give 11x
Put that at the top above the x 


               3x² + 11x 
       ---------------------- 
2x - 1 )6x³ + 19x² +   x - 10
        6x³ -  3x²
        ----------
              22x² +   x
                 
Multiply the 11x by the 2x - 1 to get 22x² - 11x,
so write that at the bottom and draw a line:

               3x² + 11x +  6
       ---------------------- 
2x - 1 )6x³ + 19x² +   x - 10
        6x³ -  3x²
        ----------
              22x² +   x
              22x² - 11x
              ----------

Now subtract, by mentally changing the signs on
the bottom row.  We get 12x, so we write that
at the bottom, and bring down the next term -10:



               3x² + 11x 
       ---------------------- 
2x - 1 )6x³ + 19x² +   x - 10
        6x³ -  3x²
        ----------
              22x² +   x
              22x² - 11x
              ----------
                     12x - 10
       
2x divides into 12x to give 6
Put that at the top above the -10 


               3x² + 11x +  6
       ---------------------- 
2x - 1 )6x³ + 19x² +   x - 10
        6x³ -  3x²
        ----------
              22x² +   x
              22x² - 11x
              ----------
                     12x - 10
                    
Multiply the 6 by the 2x - 1 to get 12x - 6,
so write that at the bottom and draw a line:


               3x² + 11x +  6
       ---------------------- 
2x - 1 )6x³ + 19x² +   x - 10
        6x³ -  3x²
        ----------
              22x² +   x
              22x² - 11x
              ----------
                     12x - 10
                     12x -  6
                     --------
                         
Now subtract, by mentally changing the signs on
the bottom row.  We get -4, so we write that
at the bottom, and we're finished because there
are no more terms to bring down:

               3x² + 11x +  6
       ---------------------- 
2x - 1 )6x³ + 19x² +   x - 10
        6x³ -  3x²
        ----------
              22x² +   x
              22x² - 11x
              ----------
                     12x - 10
                     12x -  6
                     --------
                         -  4

Now we write the answer using

   

   

or if you like:


   

Edwin