The first term in the quotient is #2x^3#:
#color(white)( (2x^2+0x+3)/color(black)(2x^2+0x+3))(2x^3color(white)(6x^4+0x^3+5x^2-x-10))/(")" color(white)(x)4x^5+6x^4+0x^3+5x^2-x-10)#
#color(white)(......................)ul(-4x^5+0x^4-6x^3)#
#color(white)(...................................)6x^4-6x^3+5x^2#
The next term in the quotient is #3x^2#:
#color(white)( (2x^2+0x+3)/color(black)(2x^2+0x+3))(2x^3+3x^2color(white)(0x^3+5x^2-x-10))/(")" color(white)(x)4x^5+6x^4+0x^3+5x^2-x-10)#
#color(white)(......................)ul(-4x^5+0x^4-6x^3)#
#color(white)(...................................)6x^4-6x^3+5x^2#
#color(white)(................................)ul(-6x^4+0x^3-9x^2)#
#color(white)(.........................................)-6x^3-4x^2-x#
The next term in the quotient is #-3x#:
#color(white)( (2x^2+0x+3)/color(black)(2x^2+0x+3))(2x^3+3x^2-3xcolor(white)(5x^2-x-10))/(")" color(white)(x)4x^5+6x^4+0x^3+5x^2-x-10)#
#color(white)(......................)ul(-4x^5+0x^4-6x^3)#
#color(white)(...................................)6x^4-6x^3+5x^2#
#color(white)(................................)ul(-6x^4+0x^3-9x^2)#
#color(white)(.........................................)-6x^3-4x^2-x#
#color(white)(...........................................)ul(+6x^3+0x^2+9x)#
#color(white)(....................................................)-4x^2+8x-10#
The last term in the quotient is #-2#:
#color(white)( (2x^2+0x+3)/color(black)(2x^2+0x+3))(2x^3+3x^2-3x-2color(white)(x-10))/(")" color(white)(x)4x^5+6x^4+0x^3+5x^2-x-10)#
#color(white)(......................)ul(-4x^5+0x^4-6x^3)#
#color(white)(...................................)6x^4-6x^3+5x^2#
#color(white)(................................)ul(-6x^4+0x^3-9x^2)#
#color(white)(.........................................)-6x^3-4x^2-x#
#color(white)(...........................................)ul(+6x^3+0x^2+9x)#
#color(white)(....................................................)-4x^2+8x-10#
#color(white)(......................................................)ul(+4x^2+0x+6)#
#color(white)(..................................................................)8x-4#
The quotient is #2x^3+3x^2-3x-2# with a remainder of #8x-4#
