What is the regular expression for the following language?
I claim that $L$ is equal to the set of all strings of the form $a\gamma b\gamma$, where $a,b \in \{0,1\}^+$ and $\gamma \in \{0,1\}$. Indeed, a string of this form clearly belongs to $L$. Conversely, if $acbc \in L$ and $c=d\gamma$, then $acbc = ad\gamma bd\gamma$ belongs to the new set. The new set is clearly described by the regular expression
$$ (0+1)^+0(0+1)^+0 + (0+1)^+1(0+1)^+1. $$