To divide −x4+6x3+8x+12 by x2−3x+9,
as −x4x2=−x2.
Now −x2×(x2−3x+9)=−x4+3x3−9x2
Subtracting −x4+3x3−9x2 from −x4+6x3+8x+12, we get
(−x4+6x3+8x+12)−(−x4+3x3−9x2)=3x3+9x2+8x+12
Now in similar way 3x3+9x2+8x+12, x2−3x+9 can go 3x times and remainder will be
3x3+9x2+8x+12−3x(x2−3x+9)
= 3x3+9x2+8x+12−3x3+9x2−27x
= 18x2−19x+12
Now in this, x2−3x+9 goes 18 times and remainder is
18x2−19x+12−18(x2−3x+9)=−19x+54x+12−162=35x−150
Hence when we divide −x4+6x3+8x+12 by x2−3x+9, the quotient is −x2+3x+18 and remainder is 35x−150
