Eine schnelle Lösung hilft uns, ruhig zu bleiben.

Because Lösung is a feminine noun: die Lösung.
So the indefinite article is eine, and the adjective ending in nominative feminine is -e:

• eine schnelle Lösung (nom. fem.)

If it were a neuter noun (like das Problem), you’d get:

• ein schnelles Problem (nom. neut.)

It’s nominative, because it’s the subject of the sentence—the thing doing the “helping.”
The verb is hilft (from helfen), so “X helps (us) ...” → X is the subject.

uns is the dative form of wir (we). The verb helfen takes a dative object (it “helps” to someone).
So:

• helfen + Dativ → jemandem helfen
• uns = dative “to us”

Not:

• wir (nominative) = “we” as a subject

uns can be accusative or dative (same form). You identify the case by the verb or structure.
Here the verb helfen requires dative, so uns must be dative.

Compare:

• Er sieht uns. (sehen takes accusative) → uns = accusative
• Er hilft uns. (helfen takes dative) → uns = dative

After helfen, German typically uses an infinitive with zu to express “help (to) do something”:

• jemandem helfen, etwas zu tun

So:

• hilft uns, ruhig zu bleiben = helps us to stay calm

You can’t say hilft uns, bleiben because zu is required in this construction.

Because ruhig here isn’t an adjective modifying a noun; it describes a state connected to the verb bleiben.
With sein/werden/bleiben, German often uses an adjective as a predicate adjective, and it doesn’t take endings:

• Wir bleiben ruhig. (not ruhige/ruhigen)

Adjective endings show up when an adjective comes before a noun:

• ein ruhiger Mensch
• die ruhige Stimme

The comma separates the main clause from the infinitive phrase (Infinitivgruppe) introduced by zu.
With zu + infinitive phrases, a comma is very common and often recommended—especially when it improves readability or when there’s a reference word like dabei, darum, etc. In practice, you’ll usually see the comma here.

You might see it without a comma in informal writing, but the comma is the standard and clearer choice. In careful writing, keep the comma:

• ..., ruhig zu bleiben.

It usually means “a quick solution” in a general sense (not necessarily a particular one already identified). Context can make it specific, but on its own it’s often generic: a quick solution (any quick solution) helps us stay calm.

Because the subject eine schnelle Lösung is singular. German verbs agree with the subject in person and number:

• Eine Lösung hilft. (singular)
• Schnelle Lösungen helfen. (plural)

Yes. A common alternative uses damit (“so that”) with a subordinate clause:

• Eine schnelle Lösung hilft uns, damit wir ruhig bleiben.

But the original infinitive construction is very natural and often more concise:

• ..., ruhig zu bleiben.