Set up a table with the coefficients of the dividend on the first row (#color(red)("row A")# below) leaving a little extra space at the beginning of the line.
Leave a blank line (to be filled in as we go along).
Write the negative of the divisor at the far left of the third line (#color(red)("row C")# below).
You should have something that looks like:
#color(white)("XXXX")color(cyan)(x^3)color(white)("XX")color(cyan)(x^2)color(white)("XX")color(cyan)(x^1)color(white)("XX")color(cyan)(x^0)#
#color(white)("XXx")|color(white)("XX")1color(white)("XX")9color(white)("XX")23color(white)("XX")15color(white)("X")color(red)(" row A")#
#underline(color(white)("xxx")|color(white)("XXXXXXXXXXXX"))color(red)(" row B")#
#-5color(white)("x")|color(white)("XXXXXXXXXXXXXX")color(red)("row C")#
For each coefficient column in turn from left to right:
Add the value in #color(red)("row B")# to the (coefficient) value in #color(red)("row A")# and write the result in #color(red)("row C")#
Multiply the negative divisor (on the far left of #color(red)("row C")#) by the value just written in #color(red)("row C")# and writer the product in the next column of #color(red)("row B")#
After processing the first column, your result should look like:
#color(white)("XXx")|color(white)("XX")1color(white)("XX")9color(white)("XX")23color(white)("XX")15color(white)("X")color(red)(" row A")#
#underline(color(white)("XXx")|color(white)("XXxx")-5color(white)("XXXXXXX"))color(red)(" row B")#
#-5color(white)("x")|color(white)("X")1color(white)("XXXXXXXXXXXx")color(red)("row C")#
...and after processing all the columns, like:
#color(white)("XXx")|color(white)("XX")1color(white)("XX")9color(white)("XX")23color(white)("XX")15color(white)("X")color(red)(" row A")#
#underline(color(white)("XXx")|color(white)("XXxx")-5color(white)("x")-20color(white)("x")-15)color(red)(" row B")#
#-5color(white)("x")|color(white)("X")1color(white)("XX")4color(white)("XXX")3color(white)("XX")0color(white)("X")color(red)("row C")#
The final result in #color(red)("row C")# gives the coefficients of the resulting quotient plus (as the last value in #color(red)("row C")# the remainder:
#-5color(white)("x")|color(white)("X")color(cyan)(underline(1color(white)("XX")4color(white)("XXX")3))color(white)("XX")color(brown)(0)color(white)("XXXXXXX")color(red)("row C")#
#color(white)("XXXX")color(cyan)(x^2color(white)("XX")x^1color(white)("XX")x^0)color(white)("XX")color(brown)("Remainder")#
