Solving $|2^{x+1}-1|+|2^{x+1}+1|=2^{|x+1|}$. How do we work on absolute value of a power?

Question

I'm trying to figure out how do we work on absolute value of a power. Like in this question.

$$|2^{x+1}-1|+|2^{x+1}+1|=2^{|x+1|}$$

What am I supposed to to do here?

I need to solve for $x$, but I'm quite not sure of what should be my approach here?

A little help would be great.