#(-3x^3-3x^2-4x+1)/(3x-4)#
In long division form:
#color(white)( (3x-4)/color(black)(3x-4)) color(white)( (-3x^3-3x^2-4x+1))/(") "-3x^3-3x^2-4x+1)#
Write #-x^2# in the quotient:
#color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2color(white)(-3x^2-4x+1))/(") "-3x^3-3x^2-4x+1)#
Multiply #-x^2(3x-4) = -3x^2+ 4x^2# and subtract from the dividend:
#color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2color(white)(-3x^2-4x+1))/(") "-3x^3-3x^2-4x+1)#
#color(white)(".....................")ul(3x^2- 4x^2)#
#color(white)(".................... .")-7x^2-4x#
Write #-7/3x# in the quotient:
#color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2-7/3xcolor(white)(-4x+1))/(") "-3x^3-3x^2-4x+1)#
#color(white)(".....................")ul(3x^2- 4x^2)#
#color(white)(".................... .")-7x^2-4x#
Multiply #-7/3x(3x-4) = -7x^2+ 28/3x# and subtract from the dividend:
#color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2-7/3xcolor(white)(-4x+1))/(") "-3x^3-3x^2-4x+1)#
#color(white)(".....................")ul(3x^2- 4x^2)#
#color(white)(".................... .")-7x^2-4x#
#color(white)("....................... .")ul(7x^2-28/3x)#
#color(white)("............................... .")-40/3x+ 1#
Write #-40/9# in the quotient:
#color(white)( (3x-4)/color(black)(3x-4)) (color(white)("...")-x^2-7/3x-40/9color(white)(1))/(") "-3x^3-3x^2-4x+1)#
#color(white)(".....................")ul(3x^2- 4x^2)#
#color(white)(".................... .")-7x^2-4x#
#color(white)("....................... .")ul(7x^2-28/3x)#
#color(white)("............................... .")-40/3x+ 1#
Multiply #-40/9(3x-4) = -40/9x+ 160/9# and subtract from the dividend:
#color(white)( (3x-4)/color(black)(3x-4)) (color(white)("...")-x^2-7/3x-40/9color(white)(1))/(") "-3x^3-3x^2-4x+1)#
#color(white)(".....................")ul(3x^2- 4x^2)#
#color(white)(".................... .")-7x^2-4x#
#color(white)("....................... .")ul(7x^2-28/3x)#
#color(white)("............................... .")-40/3x+ 1#
#color(white)("............................... .")ul(40/9x - 160/9)#
#color(white)("........................................... .")-151/9#
The quotient is #-x^2-7/3x-40/9# with a remainder of #-151/9#
