In this post, the top answer did the following

$$ \begin{split} (1+|\xi|^2)^p &\leq (1+2|\xi-\eta|^2+2|\eta|^2)^p\\ &\leq 2^p(1+|\xi-\eta|^2+1+|\eta|^2)^p\\ &\leq c(1+|\xi-\eta|^2)^p + c(1+|\eta|^2)^p, \end{split} $$ for $p>0$, where $c=\max\{2^{p},2^{2p-1}\}$.

I am not sure how was the final inequality established. Am I missing some important identity to see this inequality?

P.S. I would usually just ask this in the comment but since that question has been answered about to a decade ago, I figured it'd be better if I ask in a separate post