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)))x−2x−2−x4−4x3−9x2−7x−7)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯−x4−4x3−9x2−7x−7
We multiply xx by -x^3−x3 to get to -x^4−x4, 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)))x−2x−2−x400−x3−9x2−7x−7)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯−x4−4x3−9x2−7x−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)))x−2x−2(−x4+2x3−9x2−7x−7¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x4−6x3−9x2−7x−7)
We multiply xx by -6x^2−6x2 to get to -6x^3−6x3 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)))x−2x−2−x400−x3−6x2−7x−7)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯−x4−4x3−9x2−7x−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)))x−2x−2(−x4+2x3−9x2−7x−7¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x4−6x3−9x2−7x−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)))x−2x−2(−x4−6x3+12x2−7x−7¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x4−0x3−21x2−7x−7)
We multiply xx by -21x−21x to get to -21x^2−21x2 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)))x−2x−2−x400−x3−6x2−21x−7)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯−x4−4x3−9x2−7x−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)))x−2x−2(−x4+2x3−9x2−7x−7¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x4−6x3−9x2−7x−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)))x−2x−2(−x4−6x3+12x2−7x−7¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x4−0x3−21x2−7x−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))))x−2x−2(−x4−2x3−21x2+42x−7¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x4−0x3−0x2−49x−7)
We multiply xx by -49−49 to get to -49x−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)))x−2x−2−x400−x3−6x2−21x−49)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯−x4−4x3−9x2−7x−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)))x−2x−2(−x4+2x3−9x2−7x−7¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x4−6x3−9x2−7x−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)))x−2x−2(−x4−6x3+12x2−7x−7¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x4−0x3−21x2−7x−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))))x−2x−2(−x4−2x3−21x2+42x−7¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x4−0x3−0x2−49x−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))))x−2x−2(−x4−2x3−1x2−49x+98¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0x4−0x3−0x2−0x−105)
And so (-x^4-4^3-9x^2-7x-7)/(x-2)=-x^3-6x^2-21x-49−x4−43−9x2−7x−7x−2=−x3−6x2−21x−49 with a remainder of -105−105
