Nem bérelünk kenut anélkül, hogy előbb megnéznénk a mentőmellényeket.

Anélkül, hogy is a very common Hungarian pattern meaning without doing something or without it being the case that....

• anélkül = without that / without it
• hogy = that, introducing a clause

So anélkül, hogy előbb megnéznénk a mentőmellényeket means without first checking the life jackets.

Hungarian usually uses this full two-part structure when a whole clause follows. It is not just anélkül by itself.

After anélkül, hogy, Hungarian normally uses the conditional form. So megnéznénk is exactly the form you expect here.

In this sentence, though, it does not sound especially hypothetical in English. It is simply the standard grammar after anélkül, hogy. So the natural English meaning is still just:

without first checking...

not necessarily without first would check...

Grammatically, megnéznénk is:

• meg-néz = check / look at
• -né- = conditional marker
• -nk = we

The verb nézni means to look at / to watch.
Adding meg- often gives it a more completed or result-focused sense.

So:

• nézni = to look at, to watch
• megnézni = to look at something properly, to check it, to inspect it, to take a look at it

In this sentence, megnéznénk a mentőmellényeket suggests checking/inspecting the life jackets, not just casually looking in their direction.

Because the object is kenut, with no definite article. That makes it an indefinite object, so Hungarian uses the indefinite conjugation of the verb:

• bérelünk = we rent / we are renting with an indefinite object
• béreljük = we rent it / we rent the specific one with a definite object

Here, kenut means a canoe in a general sense, not the canoe. So bérelünk is the correct choice.

Compare:

• Nem bérelünk kenut... = We don't rent a canoe...
• Nem béreljük a kenut... = We don't rent the canoe...

Because kenut is the direct object of the verb bérelünk, and Hungarian marks direct objects with the accusative ending, usually -t.

So:

• kenu = canoe
• kenut = canoe as a direct object

This is one of the most basic case endings in Hungarian.

There are two things happening here:

1. mentőmellények = life jackets
   This is the plural form.

2. mentőmellényeket = life jackets as a direct object
   This adds the accusative ending to the plural form.

So the breakdown is:

• mentőmellény = life jacket
• mentőmellények = life jackets
• mentőmellényeket = life jackets (object)

As for a: Hungarian uses a / az as the definite article for both singular and plural nouns.

So:

• a könyv = the book
• a könyvek = the books

That is why a mentőmellényeket means the life jackets.

Here előbb means first or beforehand.

So:

• előbb megnéznénk = we would first check
• natural English: first checking / checking first

It tells you the order of actions: checking the life jackets happens before renting the canoe.

The given word order is a natural, neutral one:

Nem bérelünk kenut anélkül, hogy előbb megnéznénk a mentőmellényeket.

A few points:

• Nem usually goes directly before the verb it negates: nem bérelünk
• előbb comes before megnéznénk because it modifies that action
• the anélkül, hogy clause comes after the main clause as a normal without... phrase

Hungarian word order is flexible, but changing it usually changes focus or emphasis. For example, moving kenut earlier would put more emphasis on canoe. The version you were given is the most straightforward neutral wording.