Is there a difference between $\sqrt{x+2}+x=0$ and $x^2-x-2=0$ Is there a difference between $\sqrt{x+2}+x=0$ and $x^2-x-2=0$
Solutions are $x=2$ or $x=-1$.
But $x=2$ does not satisfy $\sqrt{x+2}+x=0$.Because $\sqrt{4}+2 \neq0$
So does it mean that they are different ? Why ?

Yes. By convention $\sqrt{x+2}$ is non-negative.
You can multiply the first by $\sqrt{x+2}-x$ to get $x+2-x^2=0$. You introduced the solution $x=2$ when you assumed $\sqrt{x+2}\sqrt{x+2}=x+2$. This is not true, since $x+2$ can be negative.
Note that this does not violate the fundamental theorem of algebra since you did not begin with a polynomial.
