It is important to insert all of the powers of x that have a 0 coefficient into the dividend.
#color(white)( (2x+1)/color(black)(2x+1))color(white)((8x^4+0x^3+0x^2+0x-4))/(")" color(white)(x)8x^4+0x^3+0x^2+0x-4)#
Please observe that the first term in the quotient is #4x^3#, because #4x^3(2x+1) = 8x^4+4x^3# and we must subtract that from the dividend:
#color(white)( (2x+1)/color(black)(2x+1))(4x^3color(white)(0x^3+0x^2+0x-4))/(")" color(white)(x)8x^4+0x^3+0x^2+0x-4)#
#color(white)(...........)ul(-8x^4-4x^3)#
#color(white)(....................)-4x^3+0x^2#
Please observe that the next term in the quotient is #-2x^2#, because #-2x^2(2x+1) = -4x^3-2x^2# and we must subtract that from the dividend:
#color(white)( (2x+1)/color(black)(2x+1))(4x^3-2x^2color(white)(0x^2+0x-4))/(")" color(white)(x)8x^4+0x^3+0x^2+0x-4)#
#color(white)(...........)ul(-8x^4-4x^3)#
#color(white)(....................)-4x^3+0x^2#
#color(white)(......................)ul(+4x^3+2x^2)#
#color(white)(....................................)2x^2+0x#
Please observe that the next term in the quotient is #x#, because #x(2x+1) = 2x^2+x# and we must subtract that from the dividend:
#color(white)( (2x+1)/color(black)(2x+1))(4x^3-2x^2+xcolor(white)(0x-4))/(")" color(white)(x)8x^4+0x^3+0x^2+0x-4)#
#color(white)(...........)ul(-8x^4-4x^3)#
#color(white)(....................)-4x^3+0x^2#
#color(white)(......................)ul(+4x^3+2x^2)#
#color(white)(....................................)2x^2+0x#
#color(white)(................................)ul(-2x^2-x)#
#color(white)(..........................................)-x-4#
Please observe that the last term in the quotient is #-1/2#, because #-1/2(2x+1) = -x-1/2# and we must subtract that from the dividend:
#color(white)( (2x+1)/color(black)(2x+1))(4x^3-2x^2+x-1/2color(white)(-4))/(")" color(white)(x)8x^4+0x^3+0x^2+0x-4)#
#color(white)(...........)ul(-8x^4-4x^3)#
#color(white)(....................)-4x^3+0x^2#
#color(white)(......................)ul(+4x^3+2x^2)#
#color(white)(....................................)2x^2+0x#
#color(white)(................................)ul(-2x^2-x)#
#color(white)(..........................................)-x-4#
#color(white)(..........................................)ul(+x+1/2)#
#color(white)(................................................)-7/2#
