Brokkoli und Blumenkohl brauchen nicht lange im Topf.

Because the subject is Brokkoli und Blumenkohl = broccoli and cauliflower.

Two nouns joined by und usually make a plural subject, so the verb must also be plural:

• der Brokkoli braucht ... = the broccoli needs ...
• der Blumenkohl braucht ... = the cauliflower needs ...
• Brokkoli und Blumenkohl brauchen ... = broccoli and cauliflower need ...

So brauchen is the correct 3rd person plural form.

In German, food items and substances are often used without an article when talking about them in a general sense, much like English:

• Brokkoli ist gesund.
• Blumenkohl schmeckt gut.

Here, the sentence is talking about those vegetables generally, not about one specific broccoli or one specific cauliflower head.

You could use articles in other contexts, for example:

• Der Brokkoli im Kühlschrank ist noch frisch.
  = The broccoli in the fridge is still fresh.

But in your sentence, article-less nouns sound natural and general.

Grammatically, each word is treated as singular on its own, but together they form a plural idea because of und.

So:

• Brokkoli → singular
• Blumenkohl → singular
• Brokkoli und Blumenkohl → plural as a combined subject

This is why the verb is plural: brauchen.

Nicht lange means not long or more naturally in English, not for long.

In this sentence, it means they do not need much time in the pot.

So the idea is:

• lange = for a long time
• nicht lange = not for a long time

This is a common German way to talk about duration:

• Das dauert nicht lange. = That won’t take long.
• Die Kartoffeln brauchen nicht lange. = The potatoes don’t need long.

Because nicht is negating the adverb lange.

The sentence is not saying the vegetables are not needed; it is saying the time is not long.

So the structure is:

• lange = long
• nicht lange = not long

This placement is very natural in German when nicht negates an adjective or adverb directly.

Compare:

• Er arbeitet nicht schnell. = He doesn’t work quickly.
• Das ist nicht schwer. = That is not difficult.

Here:

• Sie brauchen nicht lange im Topf. = They don’t need long in the pot.

Im Topf means in the pot.

Im is a contraction of:

• in dem → im

This contraction is extremely common in German.

Examples:

• im Haus = in the house
• im Garten = in the garden
• im Topf = in the pot

So:

• in dem Topf is possible
• im Topf is the normal, natural form

Because this sentence describes location, not movement into something.

German uses different cases after in:

• Dative for location: in dem Topf / im Topf = in the pot
• Accusative for direction: in den Topf = into the pot

In your sentence, the vegetables are simply in the pot while cooking, so German uses the dative:

• im Topf

Compare:

• Das Gemüse ist im Topf. = The vegetables are in the pot.
• Ich lege das Gemüse in den Topf. = I put the vegetables into the pot.

Usually, yes. Topf generally means pot, especially a cooking pot in kitchen contexts.

In this sentence, im Topf clearly means in the cooking pot / saucepan.

A few related kitchen words:

• der Topf = pot
• die Pfanne = pan
• der Kochtopf = cooking pot

In everyday German, Topf alone is often enough when the context is cooking.

Here, brauchen is a normal main verb meaning to need.

Structure:

• Brokkoli und Blumenkohl = subject
• brauchen = need
• nicht lange = not long
• im Topf = in the pot

So the sentence literally works like:

Broccoli and cauliflower need not long in the pot.

Natural English would be:

Broccoli and cauliflower don’t need long in the pot.

This is different from another common German pattern:

• Du brauchst nicht zu warten. = You don’t need to wait.

There, brauchen is followed by zu + infinitive.
In your sentence, it is simply taking a time expression: nicht lange.

Yes, that is also possible.

German often allows some flexibility with adverbial phrases such as im Topf and nicht lange.

Both of these sound natural:

• Brokkoli und Blumenkohl brauchen nicht lange im Topf.
• Brokkoli und Blumenkohl brauchen im Topf nicht lange.

The meaning stays basically the same. The difference is mostly about emphasis or rhythm.

The original version sounds very natural and smooth.

Because they are nouns, and all nouns in German are capitalized.

So:

• Brokkoli = noun
• Blumenkohl = noun
• Topf = noun

But words like und, brauchen, nicht, and lange are not nouns, so they are not capitalized.

This is one of the most noticeable spelling rules in German.

Not as a normal complete sentence on its own.

German, like English, usually needs the subject to be stated:

• Brokkoli und Blumenkohl brauchen nicht lange im Topf.

If the subject is already clear from context, you could replace it with a pronoun:

• Sie brauchen nicht lange im Topf.
  = They don’t need long in the pot.

But just Brauchen nicht lange im Topf would usually sound incomplete.

Usually, yes, or at least about cooking them in a pot of water.

The phrase im Topf strongly suggests cooking in a pot, often boiling or simmering. It tells you the cooking method rather than just the general idea of cooking.

If you wanted other cooking methods, German would normally use different phrases, for example:

• im Ofen = in the oven
• in der Pfanne = in the pan
• gedämpft = steamed

So im Topf gives a clear kitchen context.