First we solve x-1=0 to find the root, so x=1. Now we use synthetic division using this root. But first we arrange the dividend in reverse order of powers: -1x5+0x4+0x3+0x2+1x, and the "dividend" is simply the coefficients: -1 0 0 0 1. We have:
1 | -1 0 0 0 1 where "|" just separates the divisor from the dividend.
Next, we copy the lead coefficient -1 and place it beneath the -1 in the dividend and also into the "quotient" box underneath.
1 | -1 0 0 0 1
-1
-1
Taking the quotient -1 we multiply by the root 1 to give us -1 and we place this next to the underscored -1 lining it up with the zero above it:
1 | -1 0 0 0 1
-1 -1
-1 -1
We add 0+(-1)=-1 into the quotient box, and we repeat the multiplication and division until we reach the 1 in the dividend:
1 | -1 0 0 0 1
-1 -1 -1 -1 -1
-1 -1 -1 -1 | 0
The last underscored -1 is added to the 1 in the dividend: 1+(-1)=0 giving us the exact remainder 0, which I've separated from the rest of the quotient box using the vertical bar.
We have -1 -1 -1 -1 as the quotient representing the true answer in descending powers of x: -x4-x3-x2-x.
So (x-1)(-x4-x3-x2-x)=-x5-x4-x3-x2+x4+x3+x2+x=-x5+x=x-x5 is a quick check to show we got the sum right.