DS:For all nonzero integers n,n*=(n+2)/n.What is value of x

[neetigupta.86]

Q. For all non zero integers n,n*=(n+2)/n. What is the value of x ?
1. x* = x.
2. x* = -2-x.

Answer = C.

For statement 2 , the solution says
=>(x+2)/x= -2-x.
=> Multiply both sides by x.
=> (x+2) = -x(x+2)
=> x+2= -x^2 - 2x
=>x^2 + 3x + 2 = 0.
(x+1)(x+2) = 0
x = -1 . x = -2

My question is why cant we cancel out the (x+2) on LHS and RHS instead of multiplying it by x in step.
We will get x = -1.
So answer should be B instead of C.

Why cant we reduce the equation ? Shall we never do it in GMAT ?
Could anyone please explain where equations should be reduced and where they shouldn't be

[tim]

"Canceling" is not a mathematical operation. It is just shorthand for dividing both sides of an equation by the thing that is being "canceled". But remember, you can't divide by an expression that could potentially be zero, and x+2 in this case could certainly be zero if x=-2. The bottom line is you should eliminate "canceling" from your vocabulary and instead make sure everything you do with an equation is an actual mathematical operation.