Performing polynomial long division is much like regular long division. We write the polynomial and its divisor down in long division form and work through the normal steps of long division. Then we guess our first term in the quotient, which should subtract from the first term in the dividend shown in #color(red)"red"# below:
#{:("",{:("",color(red)(x^2),color(blue)(-5x),color(green)(+11)):}),(x+2,bar(")"{:(x^3,-3x^2,+x,+2):})),(,{:(color(red)(-x^3 -2x^2),):}),(,{:(bar(" "-5x^2),+x,):}),(,{:(" ",color(blue)(5x^2),color(blue)(+10x),):}),(,{:(" ",bar(" "11x),+2):}),(,{:(" "," ",color(green)(-11x),color(green)(-22)):}),(,{:(" ",bar(" "-20)):}):}#
We then repeat this step for the next powers of #x# shown in #color(blue)"blue"# and #color(green)"green"#. We end up with a remainder, which we treat in the usual way, by dividing it by the divisor and adding it to the quotient:
#(x^3-3x^2+x+2)/(x+2)=x^2-5x+11-20/(x+2)#
