Start with dividing the term with the highest exponent by the term with the highest exponent, so here, divide #3x^4# by #x#.
#3x^4 -: x = 3x^3#
Multiply #3x^3# with #x+2#: #3x^3(x+2) = 3x^4 + 6 x^3#.
Your #x^3# term is #2x^3#, so to compute your remainder for the #x^3# term, compute #2x^3 - 6x^3 = - 4 x^3#.
So, you can already transform your numerator as follows:
#3x^4 + 2x^3 - x^2 + 2x - 9 = color(green)(3x^3(x+2)) color(red)( - 4x^3 - x^2 + 2x - 9)#.
The green coloured terms are already done, now take care of the red coloured terms.
Again, divide the term with the highest exponent:
#(-4x^3) -: x = -4 x^2#
Multiply with #x+2#:
#-4 x^2 (x+2) = - 4 x^3 - 8 x^2#
The remainder for your #x^2# term is:
#- x^2 - (-8 x ^2) = 7 x^2#.
So,
#3x^4 + 2x^3 - x^2 + 2x - 9#
#= color(green)(3x^3(x+2)) - 4x^3 - x^2 + 2x - 9#
# = color(green)(3x^3(x+2) - 4x^2(x+2)) color(red)( + 7x^2 + 2x - 9).#
Repeate the procedure with the remaining red term.
Divide: #7x^2 -: x = 7x#
Multiply: #7x (x+2) = 7x^2 + 14x#
Remainder: #2x - 14x =- 12x#
In total:
#3x^4 + 2x^3 - x^2 + 2x - 9#
#= color(green)(3x^3(x+2)) - 4x^3 - x^2 + 2x - 9#
# = color(green)(3x^3(x+2) - 4x^2(x+2)) + 7x^2 + 2x - 9#
# = color(green)(3x^3(x+2) - 4x^2(x+2) + 7x( x + 2)) color(red)(- 12x - 9).#
Last step: try to divide #-12x - 9# by #x + 2#:
Divide: #-12x -: x = -12#
Multiply: #-12(x+2) = -12 x - 24#
Remainder: #-9 - (-24) = 15#
As the degree of the remainder (#0#, since no #x# or rather #x^0#) is lower than the degree of the denominator (degree #1#, highest exponent #x^1#), you can stop here.
The result is:
#3x^4 + 2x^3 - x^2 + 2x - 9#
#= color(green)(3x^3(x+2)) - 4x^3 - x^2 + 2x - 9#
# = color(green)(3x^3(x+2) - 4x^2(x+2)) + 7x^2 + 2x - 9#
# = color(green)(3x^3(x+2) - 4x^2(x+2) + 7x( x + 2)) - 12x - 9#
# = color(green)(3x^3(x+2) - 4x^2(x+2) + 7x( x + 2) - 12(x + 2)) + 15#
So, the result of
#(3x^4 + 2x^3 - x^2 + 2x - 9) -: (x+2)# is
#3x^3 - 4x^2 + 7x - 12# with the remainder #15#, or:
#(3x^4 + 2x^3 - x^2 + 2x - 9) / (x+2) = 3x^3 - 4x^2 + 7x - 12 + 15/(x+2)#
Hope that this helped!
