We can do long division on this:
#color(white)((x-2)/color(black)(x-2)(-x^4-4x^3-9x^2-7x-7)/color(black)(")"bar(-x^4-4x^3-9x^2-7x-7)))#
We multiply #x# by #-x^3# to get to #-x^4#, and so we have:
#color(white)((x-2)/color(black)(x-2)(-x^4color(black)(color(white)(00)-x^3)-9x^2-7x-7)/color(black)(")"bar(-x^4-4x^3-9x^2-7x-7)))#
#color(white)((x-2)/(x-2)((color(black)(-x^4+2x^3)-9x^2-7x-7)/(color(black)bar(0x^4-6x^3-9x^2)-7x-7)))#
We multiply #x# by #-6x^2# to get to #-6x^3# and so we have:
#color(white)((x-2)/color(black)(x-2)(-x^4color(black)(color(white)(00)-x^3-6x^2)-7x-7)/color(black)(")"bar(-x^4-4x^3-9x^2-7x-7)))#
#color(white)((x-2)/(x-2)((color(black)(-x^4+2x^3)-9x^2-7x-7)/(color(black)bar(0x^4-6x^3-9x^2)-7x-7)))#
#color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4)-6x^3+12x^2)-7x-7)/(color(black)bar(color(white)(0x^4-)0x^3-21x^2-7x)-7)))#
We multiply #x# by #-21x# to get to #-21x^2# and so we have:
#color(white)((x-2)/color(black)(x-2)(-x^4color(black)(color(white)(00)-x^3-6x^2-21x)-7)/color(black)(")"bar(-x^4-4x^3-9x^2-7x-7)))#
#color(white)((x-2)/(x-2)((color(black)(-x^4+2x^3)-9x^2-7x-7)/(color(black)bar(0x^4-6x^3-9x^2)-7x-7)))#
#color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4)-6x^3+12x^2)-7x-7)/(color(black)bar(color(white)(0x^4-)0x^3-21x^2-7x)-7)))#
#color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4-2x^3)-21x^2+42x)-7)/(color(black)bar(color(white)(0x^4-0x^3)-0x^2-49x-7))))#
We multiply #x# by #-49# to get to #-49x# and so we have:
#color(white)((x-2)/color(black)(x-2)(-x^4color(black)(color(white)(00)-x^3-6x^2-21x-49))/color(black)(")"bar(-x^4-4x^3-9x^2-7x-7)))#
#color(white)((x-2)/(x-2)((color(black)(-x^4+2x^3)-9x^2-7x-7)/(color(black)bar(0x^4-6x^3-9x^2)-7x-7)))#
#color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4)-6x^3+12x^2)-7x-7)/(color(black)bar(color(white)(0x^4-)0x^3-21x^2-7x)-7)))#
#color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4-2x^3)-21x^2+42x)-7)/(color(black)bar(color(white)(0x^4-0x^3)-0x^2-49x-7))))#
#color(white)((x-2)/(x-2)((color(black)(color(white)(-x^4-2x^3-1x^2)-49x+98))/(color(black)bar(color(white)(0x^4-0x^3-0x^2)-0x-105))))#
And so #(-x^4-4^3-9x^2-7x-7)/(x-2)=-x^3-6x^2-21x-49# with a remainder of #-105#
