#(x^2-x-12)/(x-4)#
This can be done as a "normal" division.
First note that the first term of the dividend is #x^2# and the one of divisor is #x#. If you divide one by the other, you will obtain #x#.
#x^2-x-12 | ul (x-4)#
#color(white)(..................... )x#
Then you multiply #x# by #x-4# and subtract the result to the dividend:
#color(white)(...)x^2-x-12 | ul (x-4)#
#-x^2+4x color(white)(.........) x#
#----#
#color(white)(. ..)0-3x-12#
Now, you must repeat the same with #-3x-12# which gives 3:
#color(white)(...)x^2-x-12 | ul (x-4)#
#-x^2+4x color(white)(.........) x-3#
#----#
#color(white)(. ..)0-3x-12#
#color(white)(. ..)0+3x+12#
#----#
#color(white)(. ..)0color(white)(. ..)0color(white)(. ..)0#
