Statement 1: x* = x means (x+2)/x = x
Multiply both sides by x: x + 2 = x^2
Bring terms to the side of x^2: x^2 - x - 2 = 0
Factor: (x - 2)(x + 1) = 0
[spoiler]2 values of x (2 OR -1), Insufficient[/spoiler]

Statement 2: x* = -2 - x means (x+2)/x = -2 - x
Multiply both sides by x: x + 2 = -2x - x^2
Bring terms to the left: x^2 + 3x + 2 = 0
Factor: (x + 2)(x + 1) = 0
[spoiler]2 values of x (-2 OR -1), Insufficient[/spoiler]

Combined:
St 1: [spoiler]x = 2 or x = -1[/spoiler]
St 2: [spoiler]x = -2 or x = -1[/spoiler]
[spoiler]Conclusion: x = -1. Sufficient.[/spoiler]

IMO:C

Statement 2: x* = -2 - x means (x+2)/x = -2 - x
Multiply both sides by x and taking '-' out on RHS: x + 2 = -x(2 + x)
why should we not divide by x + 2 and get x = -1 as the answer

then B alone will be sufficient

Multiply both sides by x and taking '-' out on RHS: x + 2 = -x(2 + x)
why should we not divide by x + 2 and get x = -1 as the answer

This is a good question.

The answer is that by dividing both sides by the same variable expression, you are making the assumption that this variable expression is not equal to zero.

However, it is possible that (2+x) = 0. Which means that it is possible that x = -2. You can confirm that this is a valid solution to the equation by plugging it into x + 2 = -x(2 + x).
You will get 0 = 0, which is true. And since it is a valid solution, we can't ignore it.

The lesson here: Never divide by a variable expression. Instead, bring all variable terms to one side of the equation, then simplify or factor.

In this case:
x + 2 = -x(2 + x)
x + 2 = -2x - x^2
x + 2 + 2x + x^2 = 0
x^2 + 3x + 2 = 0
(x + 2)(x + 1) = 0
So, there are two solutions for Statement 2.