Long division with equations?

You do the same steps as if you were doing integers:

Find out what goes into the highest power, multiply it through, subtract it, carry down anything unused, continue until complete:

. . . . . .__________
3x + 2 ) 12x² - x - 9

There's divide 3x by 12x², and you get 4x, so that goes up top, multiplied through 3x + 2, and then subtracted from 12x² - x - 9:

. . . . . .__4x______
3x + 2 ) 12x² - x - 9
. . . . . . .12x² + 8x
. . . . . . .-----------------
. . . . . . . . . . -9x - 9

And one more time, -3 3x's into -9x:

. . . . . .__4x_-_3__
3x + 2 ) 12x² - x - 9
. . . . . . .12x² + 8x
. . . . . . .-----------------
. . . . . . . . . . -9x - 9
. . . . . . . . . . -9x - 6
. . . . . . . . . . -----------
. . . . . . . . . . . . . .-3

-3 is our remainder, so that becomes a fraction -3/(3x + 2)

So your final answer is:

4x - 3 - 3/(3x + 2)

= (12x² - x - 9)/(3x + 2)
= 4x - 3 & - 3

Answer: 4x - 3 & - 3

Proof:
= ([4x - 3][3x + 2]) - 3
= 12x² + 8x - 9x - 6 - 3
= 12x² - x - 9

The answer is 4x -3 with remainder of -3

....................4x - 3
.........___________
3x + 2)12x^2 - x - 9
........12x^2 + 8x
-----------------------------
...................-9x - 9
...................-9x - 6
-----------------------------
.........................-3

(12x^2 - x - 9)/(3x + 2)
= (4x - 3) + (-3)/(3x + 2)
= (4x - 3) - 3/(3x + 2)

3x+2)12x^2 -x -9(4x-3
-12x^2-8x
__________
-9x -9
9x +6
___________
-3
____________

therefore the quotient is 4x-3 and the reminder is -3

answer is 4x+7/9-67/27x
4x plus 7 divided by 9 minus 67 divided by 27x

it is 4x-3 and remainder is -3