Learn Polish: Mieli pecha i nie mogli do nas dołączyć. - They had bad luck and couldn't join us.

Mogli is the past tense of móc (can/to be able to) for a group containing at least one male. The verb móc has a stem change in the past tense: it uses -ó- for the masculine singular (as in on mógł), but changes to -o- for all other forms, including ona mogła and oni mogli. If the group was entirely female, you would use one mogły.

Why do we say mieli pecha instead of "mieli pech"? Isn't pech (bad luck) a masculine inanimate noun?

You are right that pech is masculine inanimate, and usually those nouns don't change their form when they are the direct object. However, mieć pecha (to have bad luck) is a fixed, everyday phrase where the noun acts like an animate object and takes an -a ending. This is similar to how you say jeść banana (to eat a banana) instead of "jeść banan".

Why does the sentence use do nas?

The preposition do (to/towards) always requires the noun or pronoun that follows it to be in the genitive case. Nas is the genitive form of my (we), so do nas translates to "to us".

Is dołączyć (to join) perfective or imperfective?