monatelang
for months
Continuing over a period of several months.
Die Bauarbeiten an der Brücke dauerten monatelang.
The construction work on the bridge lasted for months.

for months
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for months
Continuing over a period of several months.
Die Bauarbeiten an der Brücke dauerten monatelang.
The construction work on the bridge lasted for months.

The word is made up of 'Monat' (month) and 'lang' (long). This pattern is very productive and is also used with other units of time, for example 'tagelang' (for days), 'wochenlang' (for weeks), or 'jahrelang' (for years).
Wir haben jahrelang für dieses Haus gespart.
We saved for this house for years.
As an adverb, 'monatelang' (for months) can stand flexibly in the sentence. To emphasize the duration especially strongly, it can also stand in position 1, which leads to inversion (verb in position 2).
Er hat monatelang für die Prüfung gelernt.
He studied for the exam for months.
'Monatelang' (for months) is often combined with verbs that describe an ongoing process. Common examples are 'monatelang warten' (to wait for months), 'monatelang suchen' (to search for months), 'monatelang arbeiten' (to work for months), or 'sich monatelang vorbereiten' (to prepare for months).
Die Forscher haben monatelang an dem Experiment gearbeitet.
The researchers worked on the experiment for months.
long-term
'langfristig' and 'monatelang' both refer to a longer period of time. 'langfristig' is used when something is planned for the longer-term future or is meant to have a long-term effect. 'monatelang' says more specifically that something lasts or lasted for several months. The risk of confusion comes from the fact that several months can also feel like a long time. Even so, 'langfristig' is more a word for planning and perspective, while 'monatelang' describes the actual duration.

Die Kundin hat monatelang auf einen freien Termin gewartet.
The customer waited for months for an available appointment.
Eine gute Freundschaft bleibt langfristig nur mit Vertrauen stabil.
In the long term, a good friendship only stays stable with trust.