#(color(brown)(4)x^2color(brown)(-2)xcolor(brown)(+6))div (color(cyan)(2)xcolor(blue)(-3))#

Remember to reverse the sign on #color(blue)((-3))#

and since we have a non-monic divisor, we need to divide each column sum by (in this case) #color(cyan)(2)#

"Bring down" the first coefficient

Then divide by 2

#{:
( ," | ",color(brown)(4),color(brown)(-2),color(brown)(+6)),
(color(blue)(+3)," | ", , , ),
( ," | ","----","----","----"),
(color(cyan)(/2)," | ",4,,),
(," | ",2,color(white)("X")2,)
:}#

Multiply the last column quotient (#2#) by #3# and write in the next column.

Add that column.

#{:
( ," | ",4,-2,+6),
(+3," | ", , color(white)("X")6 , ),
( ," | ","----","----","----"),
(/2," | ",4,color(white)("X")4,),
(," | ",2,,)
:}#

Repeat this process until done

#{:
( ," | ",4,-2,+6),
(+3," | ", , color(white)("X")6 , color(white)("X")6),
( ," | ","----","----","----"),
(/2," | ",4,color(white)("X")4,color(white)("X")color(red)(12)),
(," | ",color(green)(2),color(white)("X")color(orange)(2),)
:}#

The last sum (undivided), #color(red)(12)#, is the remainder.

The quotients preceding the last column, #color(green)(2)# and #color(orange)(2)#, are the coefficients of the quotient expression.

That is the solution is

#color(green)(2)x+color(orange)(2)# with a Remainder of #color(red)(12)#