Example of a non locally-compact space with a dense locally compact subspace. Let $X$ be a topological space such that $A \subset X$ is a dense subspace which is locally compact and $B \subset X$ is a dense subspace which is not locally compact (at all of its points).
Is it possible to find such $X$? if it is, an example?

There remains some imprecision in how you've worded the question, but here's a non-locally compact space with a locally compact, indeed compact, dense subset. Just take your favorite non-locally compact space $X$ and construct $X'$ by adjoining a generic point $\star$, modifying the topology by adding $\star$ to every open set, including the empty set (of course we must add the empty set back in again!) 
Then the closure of $\{\star\}$ is $X'$ but, being finite, $\{\star\}$ is compact. On the other hand, $X'$ still isn't locally compact at any of the points at which $X$ wasn't, because $X$ is closed and closed subspaces of locally compact spaces are locally compact.
