Here's what the set up looks like...
#1| 1 + 3 -1 - 3#
The #1# on the left is the "root" of the bottom, or the number that x would be to make it equal #0#. Or you can just think of it is the opposite of the number after #x#.
All of the numbers after are the coefficients in front of the powers of #x# in decreasing order.
Now the process. Draw leave some space and draw a line underneath like so...
#1| 1 + 3 -1 - 3#
# -------#
Add down and multiply up diagonally by the number in the top left (I hope this makes sense)
#1| 1 + 3 -1 - 3#
#color(white)(Xll) 0color(white)(XXXXXXXXX)##<---#start with a zero here
# -------#
#color(white)(Xll) 1color(white)(XXXXXXXXX)# #<---# add down
#1| 1 + 3 -1 - 3#
#color(white)(Xll) 0color(white)(llll) 1color(white)(XXXXXXX)##<---#multiply up by 1
# -------#
#color(white)(Xll) 1color(white)(Xl)4color(white)(XXXXXX)# #<---# add down
Here's the rest:
#1| 1 + 3 -1 - 3#
#color(white)(Xll) 0color(white)(llll) 1color(white)(llla) 4color(white)(llla) 3#
# -------#
#color(white)(Xll) 1color(white)(Xl)4color(white)(iiia) 3 color(white)(iiia)0#
Now use those numbers on the bottom as coefficients for a polynomial with degree one lower than the original numerator:
#1x^2 + 4x + 3 + 0/(x-1)# (this last term is the remainder, and this is how you write it)
Clean it up...
#x^2 + 4x + 3#
