How to solve example with many modules? How to solve example with many modules like this:  $| x^3  + x + 1| +  |x^3 - x - 1| = 2x + 1$?

Your example looks like $|a + b| + |a - b|$, where $a = x^3$ and $b = x + 1$.
So if $a > b$ then $2x^3 = 2x + 1$. The only solution (approximately 1.19) does not agree with this.
If $a \le b$ then $2x + 2 = 2x + 1$. So no real solutions.
Therefore you do not have solutions here.
