Show that NP is closed under concatenation
The easiest way to prove that NP is closed under concatenation is the following: 

Assume that $L_1 , L_2 \in NP$. Thus, there are two nondeterministic deciders $M_1$ and $M_2$ such that $M_1$ decides $L_1$ in nondeterministic time $O(n^l)$ and $M_2$ decides $L_2$ in nondeterministic time $O(n^k)$. 
Let $M$ a TM s.t. 


  $M$ on input $w$:
  
  
  Split $w$ into $w_1 \in L_1$ and $w_2 \in L_2$ s.t. $w = w_1 w_2$;
  Run $M_1$ on $w_1$. If $M_1$ rejects then reject;
  Else run $M_2$ on $w_2$, if $M_2$ rejects then reject;
  Else accept.
  


This will use $O(n^{\max{l,k}})$ steps. So $M$ is a polytime nondeterministic decider for $L_1L_2$.