Regular concat with non regular
The claim is this:


  If $L$ is a regular language and $M$ is a non-regular language, then the concatenation $LM$ of the two languages is necessarily non-regular.


If this claim is false, all you need to do is to provide one counterexample showing that it's false. If the claim is true, you need to prove that it's true for any languages $L$ and $M$, a much harder task. Fortunately, this claim is not true.

If we let $L = \emptyset$ and let $M$ be any non-regular language, then $LM = \emptyset = L$, a regular language.

Now, if that feels too much like cheating, we can find other examples. Consider this:


$L = \Sigma^*$
$M = \{ww^R \mid w \in \Sigma^*\}$


$L$ is regular, $M$ is non-regular (in fact, it's the language of palindromes, a canonical context-free language). Since the empty word is in $M$, we have $LM = \Sigma^* = L$, so the concatenation is regular.