The expression you provided is:

$\left(x - 1\right) + \left(x + 1\right)$

Now, let's simplify it step by step:

1. Combine the like terms: $(x - 1) + (x + 1) = x + x - 1 + 1$

2. Simplify further by canceling out $-1$ and $+1$: $x + x = 2x$

Thus, the simplified expression is:

$2x$

Would you like any additional details on the steps?

Here are 5 related questions to explore further:

1. What happens if we add constants to this expression, like $\left(2x - 1\right) + \left(3x + 1\right)$?
2. How do we simplify more complex algebraic expressions with different variables?
3. Can this expression be used in a system of equations? How would we solve that?
4. What if we include exponents, such as $\left(x^2 - 1\right) + \left(x^2 + 1\right)$?
5. How does this simplify if $x$ is given a specific value?

Tip: Always combine like terms (such as variables or constants) first to simplify expressions quickly.