Solution: (1) Because there is only one intersection with the X axis, the vertex is on the X axis.

The solution is m=2.

(2) Substitute y=x+2m for y=x? +2x+m- 1, and the discriminant is equal to 0.

That is x+2m = x 2+2x+m- 1.

x^2+x-m- 1=0

a= 1，b= 1，c=m- 1

Substitute b 2-4ac = 0.

m=5/4