You asked how to do it so I am explaining the process:
You divide the sequential x parts in the numerator into sequential x parts of the denominator. Each stage leaves a remainder for which the process is repeated.
Demonstration within the context of this question.
color(green)("==============================")
Step 1: 4x^3 divide x = 4x^2
Used the 4x^3 from 4x^3+2x-6 and the x from x-1
So the color(red)("first") part of your answer is color(blue)(4x^2)
color(green)("============================")
Step 2: Find the remainder
4x^2 times (x-1) = 4x^3 - 4x
This is then subtracted so we have:
4x^3 + 2x -6 .......Original equation
(4x^3 -4x) - ...... Subtract
~~~~~~~~~~~~~
color(white)("xxxxxx") 6x - 6. This is the first remainder
color(green)("=================================")
Step 3.
Again divide the 6x in the previous remainder by the x in x-1 giving 6.
So the color(red)("second")" " part of the answer is color(blue)(+6)
6(x-1) =6x-6 which is subtracted from the most recent remainder giving:
4x^3 + 2x -6 .......Original equation
(4x^3 -4x) - ...... Subtract
~~~~~~~~~~~~~
color(white)("xxxxxx") 6x - 6. This is the remainder
color(white)("xxxxxx") ( 6x - 6) -. Subtract
~~~~~~~~~~~~~~~~~~~~~~~~~~
color(white)("xxxxxxx") 0 + 0" " which is the second remainder.
The zeros mean that we have an exact division
In this case
(4x^3 +2x-6)/(x-1) =4x^2+6
Suppose we had ended up with a remainder that the x " in " (x-1) could not be divided into. In that case we would express the whole of that remainder as a fraction with (x-1) as the denominator.
Suppose that had ended up with a remainder of just 2. Then in that case the answer would be:
4x^2 + 2/(x-1) + 6
Hope this helps. It takes a lot of practice.
