Problem:  Divide  
6x2 + x - 9
2x - 1


Set up the long division
3x =   6x2
2x
.
Choose 3x since
3x · 2x matches 6x2.
6x2 - 3x = 3x(2x - 1).
Subtract 6x2 - 3x from 6x2 + x - 9.
Result is 4x - 9.
2 =   4x
2x
.
Choose 2 since
2 · 2x matches 4x.
4x - 2 = 2(2x - 1).
Subtract 4x - 2 from 4x - 9.
Result is -7.
-7 is the remainder.


Answer:   6x2 + x - 9
2x - 1
 =   3x + 2 - 7
2x - 1
.


Notes
6x2 + x - 9 = (2x - 1)(3x + 2) - 7
Dividend 6x2 + x - 9
Divisor 2x - 1
Quotient 3x + 2
Remainder -7