If the polynomial x^2+bx+c has exactly one real root and b=c+1, find the value of the product of all possible values of c.

We have that

x^2 + (c + 1)x + c = 0

If this has one real root, it is a double-root and the discriminant = 0

So.......

(c + 1)^2 - 4(1)c = 0

c^2 + 2c + 1 - 4c = 0

c^2 - 2c + 1 = 0 factor

(c - 1)^2 = 0

So.....c = 1

This is the only value of c that makes this true....