This exercise come from Sipser (1.68).
In essence, show that if $A$ is regular, then $CUT(A) := \{yxz | xyz \in A\}$ is regular. I've managed to show that $B = \{yx | xyz \in A\}$ is regular. Furthermore, I've shown that $SUFFIX(A) = \{z | xz \in A\}$ is regular. My idea was to then say that $CUT(A)$ is precisely the concatenation of $B$ and $SUFFIX(A)$. However, the trouble I'm running into is that this operation will attach the "wrong" suffixes to a string. For instance, if $A$ is "strings in alphabetical order", then $bad$ will be in the concatenation of $B$ and $SUFFIX(A)$.
How can I complete this?
