Rules for Factorisation? Basically presented with this, simplify
\begin{aligned}
{\Bigl(\sqrt{x^2 + 2x + 1}\Big)  +  \Bigl(\sqrt{x^2 - 2x + 1}\Big)} 
\end{aligned}
Possible factorisations into both
\begin{aligned}
{\Bigl({x + 1}\Big)^2}, {\Bigl({x - 1}\Big)^2} 
\end{aligned}
\begin{aligned}
{\Bigl({1 + x}\Big)^2} , {\Bigl({1 - x}\Big)^2} 
\end{aligned}
Hence when simplified, answer has two possibilities. One independent of x, and the other not.
( Simplified Answers: 2x, 2 )

Why is one independent and the other not? If such is equal to 2, why then when, say x=2, the answer does not simplify to 2?

You can write it as
$$\sqrt{x^2 + 2x + 1}  +  \sqrt{x^2 - 2x + 1}=\sqrt{(x+1)^2}  +  \sqrt{(x-1)^2}$$
But note that this simplifies to
$$|x+1|  +  |x-1|$$
as squaring and taking the root eliminates the sign. So neither of your answer is correct (for a real $x$). Maybe you can figure out the intervals where something interesting happens on your own?
