To make it easier to type I will leave out the ^ for
exponents on the right.  For example 2x3 means 2x3,
OK?

Begin with this:

               
      ---------------------- 
4x - 5)4x3 - 17x2 +  7x + 10
       
 

divide 4x into the 4x3, get x2, write it over the -17x2

               x2 
      ---------------------- 
4x - 5)4x3 - 17x2 +  7x + 10
       

multiply x2 by 4x - 5 and get 4x3 - 5x2, write that 
underneath the first two terms, and draw a line.

               x2
      ---------------------- 
4x - 5)4x3 - 17x2 +  7x + 10
       4x3 -  5x2
       ----------

Subtract by imagining the lower number to have the
opposite sign and combine them, getting 0x3 - 12x2, but
we don't write the 0x3 and we just have -12x2 to write below the line.
We bring down the next term, which is +7x:

               x2 
      ---------------------- 
4x - 5)4x3 - 17x2 +  7x + 10
       4x3 -  5x2
       ----------
           - 12x2 +  7x 

divide 4x into the -12x2, get -3x, write it at the top 
over the +7x

               x2 -  3x 
      ---------------------- 
4x - 5)4x3 - 17x2 +  7x + 10
       4x3 -  5x2
       ----------
           - 12x2 +  7x
        
multiply -3x by 4x - 5 and get -12x2 + 7x, write that 
underneath the two terms at the bottom, and draw a line
underneath.

               x2 -  3x 
      ---------------------- 
4x - 5)4x3 - 17x2 +  7x + 10
       4x3 -  5x2
       ----------
           - 12x2 +  7x
           - 12x2 + 15x 
           ------------
                 
Subtract by imagining the lower number to have the
opposite sign and combine them, getting 0x2 - 8x, but
we don't write the 0x2 and we just have -8x to write below the line.
and bring down the +10

               x2 -  3x 
      ---------------------- 
4x - 5)4x3 - 17x2 +  7x + 10
       4x3 -  5x2
       ----------
           - 12x2 +  7x
           - 12x2 + 15x 
           ------------
                  -  8x + 10


divide 4x into the -8x, get -2, write it at the top 
over the +10

               x2 -  3x -  2
      ---------------------- 
4x - 5)4x3 - 17x2 +  7x + 10
       4x3 -  5x2
       ----------
           - 12x2 +  7x
           - 12x2 + 15x 
           ------------
                  -  8x + 10
                


multiply -2 by 4x - 5 and get -8x + 10, write that 
underneath the two terms at the bottom, and draw a line
underneath.

               x2 -  3x -  2
      ---------------------- 
4x - 5)4x3 - 17x2 +  7x + 10
       4x3 -  5x2
       ----------
           - 12x2 +  7x
           - 12x2 + 15x 
           ------------
                  -  8x + 10
                  -  8x + 10
                  ---------- 
                           

Subtract by imagining the lower number to have the
opposite sign and combine them, getting 0 remainder
and we are through.  Since the remainder was 0,
the quotient is x2 - 3x - 2.

Edwin