To divide #-x^4+6x^3+8x+12# by #x^2-3x+9#,
as #-x^4/x^2=-x^2#.
Now #-x^2xx(x^2-3x+9)=-x^4+3x^3-9x^2#
Subtracting #-x^4+3x^3-9x^2# from #-x^4+6x^3+8x+12#, we get
#(-x^4+6x^3+8x+12)-(-x^4+3x^3-9x^2)=3x^3+9x^2+8x+12#
Now in similar way #3x^3+9x^2+8x+12#, #x^2-3x+9# can go #3x# times and remainder will be
#3x^3+9x^2+8x+12-3x(x^2-3x+9)#
= #3x^3+9x^2+8x+12-3x^3+9x^2-27x#
= #18x^2-19x+12#
Now in this, #x^2-3x+9# goes #18# times and remainder is
#18x^2-19x+12-18(x^2-3x+9)=-19x+54x+12-162=35x-150#
Hence when we divide #-x^4+6x^3+8x+12# by #x^2-3x+9#, the quotient is #-x^2+3x+18# and remainder is #35x-150#
