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#