ausführen
to carry out
To put a plan, an order, a job, or an activity into action; to execute something.
Die Bauarbeiten werden von einer lokalen Firma ausgeführt.
The construction work is being carried out by a local company.

to carry out / to elaborate / to export
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to carry out
To put a plan, an order, a job, or an activity into action; to execute something.
Die Bauarbeiten werden von einer lokalen Firma ausgeführt.
The construction work is being carried out by a local company.

'Ausführen' (to carry out / to elaborate / to export) is a separable verb. The prefix 'aus' goes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Der Architekt führt den Plan exakt aus.
The architect carries out the plan exactly.
The verb consists of the prefix 'aus' and the base verb 'führen' (to lead). The basic idea of bringing something out or leading something to completion connects the different meanings.
Die Firma führt die bestellten Waren morgen aus.
The company will export the ordered goods tomorrow.
In the meaning 'durchführen' (to carry out), the verb is often combined with plans, orders, or work. Typical objects are: an assignment, a plan, a repair, or an instruction.
Der Handwerker hat die Reparatur schnell ausgeführt.
The handyman carried out the repair quickly.
When it is about explaining, 'ausführen' (to elaborate) is often used with adverbs such as 'genauer' (more precisely / in more detail), 'detaillierter' (in more detail), or 'weiter' (further). It is a polite way to ask for more information.
Der Experte führte seine Gedanken zum Thema weiter aus.
The expert elaborated further on his thoughts about the topic.
In the meaning 'exportieren' (to export), 'ausführen' belongs to formal business or administrative language. It is often used in reports or official documents.
Das Unternehmen führt seit Jahren erfolgreich Maschinen nach Asien aus.
The company has been successfully exporting machines to Asia for years.
In this meaning, 'ausführen' (to take out) always requires an accusative object. The object is usually a person you take to a social event, or a pet for a walk.
Am Samstagabend hat er seine neue Freundin schick zum Essen ausgeführt.
On Saturday evening, he took his new girlfriend out for a fancy dinner.
to do / make
'machen' and 'ausführen' can both mean that you do an activity. 'machen' is very general and fits almost any everyday situation. 'ausführen' is more specific: you put a plan, assignment, order, or specific task into action. It often sounds somewhat more formal or technical than 'machen'. If you are just talking casually about an activity, you usually use 'machen'. If it is clear that someone is acting according to a plan or assignment, 'ausführen' fits better.

Die Mitarbeiterin führt den Auftrag genau nach Plan aus.
The employee carries out the assignment exactly according to plan.
Die Gruppe macht morgen eine kurze Übung für das Projekt.
The group is doing a short exercise for the project tomorrow.