A set of real numbers with the binary operation defined as a*b = a + b + a.b is not a group, because the inverse element does not exist for all elements in the set.
To show this, we need to find an inverse element a^-1 for a given element a such that a*a^-1 = e, where e is the identity element.
In this case, we have aa^-1 = a + a^-1 + aa^-1 = 0.
This equation does not have a real solution for a^-1 for all a, thus the set of real numbers with this binary operation does not form a group.