Is P closed under subwords? Given a language $L\subseteq \Sigma^*$ in $P$, is the language

$subwords(L) = \{v\in\Sigma^* : \text{there exist } u,w\in \Sigma^* \text{ with } uvw\in L\}$  

that consists of all subwords of words in $L$ also guaranteed to lie in $P$?
Hint: Let $L$ be the language consisting of a description of a Turing machine (bounded by $\#$ on both sides) followed by a terminating computation (both description and computation are over $\{0,1\}$). What is the language $\mathit{subwords}(L) \cap \#\Sigma^*\#$?