Terminamos la investigación.

Because terminamos can mean both:

• present tense: we finish
• preterite (simple past): we finished

For -ar verbs, the nosotros form is the same in those two tenses.

So:

• Terminamos la investigación. = We finish the investigation.
• Terminamos la investigación. = We finished the investigation.

Context tells you which one is meant. For example:

• Siempre terminamos la investigación a tiempo. = We always finish the investigation on time. → present
• Ayer terminamos la investigación. = Yesterday we finished the investigation. → past

Spanish often leaves out subject pronouns because the verb ending already shows who is doing the action.

• terminamos already tells you the subject is we
• so (nosotros) terminamos and terminamos both mean we finish / we finished

Including nosotros is possible, but it usually adds emphasis or contrast:

• Nosotros terminamos la investigación, no ellos.
  We finished the investigation, not them.

La is the definite article, meaning the.

So:

• la investigación = the investigation

Spanish uses articles very regularly, often more regularly than English. Here it refers to a specific investigation, so la is needed.

Compare:

• Terminamos la investigación. = We finished the investigation.
• Terminamos una investigación. = We finished an investigation.

Because investigación is a feminine noun.

A very useful pattern is that nouns ending in -ción are usually feminine:

• la investigación
• la información
• la conversación
• la situación

So the article must also be feminine:

• la investigación

Terminar is a regular -ar verb. It means to finish, to end, or to complete, depending on context.

Its present tense forms are:

• yo termino
• tú terminas
• él/ella/usted termina
• nosotros terminamos
• vosotros termináis
• ellos/ellas/ustedes terminan

Its preterite forms are:

• yo terminé
• tú terminaste
• él/ella/usted terminó
• nosotros terminamos
• vosotros terminasteis
• ellos/ellas/ustedes terminaron

Notice again that nosotros terminamos appears in both tenses.

Yes. In this sentence, la investigación is the direct object: it is the thing being finished.

• Terminamos = we finish / finished
• la investigación = the thing we finish / finished

You can test this by replacing it with a direct object pronoun:

• Terminamos la investigación.
• La terminamos. = We finished it / We finish it.

Here, la means it, referring to la investigación.

Yes. That means We have finished the investigation.

In Spain, the present perfect (hemos terminado) is very common for recent past actions, especially when they are connected to the present.

So in Spain Spanish:

• Terminamos la investigación. = We finished the investigation.
• Hemos terminado la investigación. = We have finished the investigation.

Both are correct, but they do not always sound exactly the same:

• terminamos often sounds like a completed past event
• hemos terminado often highlights the result now or a more current/recent relevance

The accent mark shows where the stress goes:

• in investigación, the stress is on the last syllable:
  in-ves-ti-ga-ción

Words ending in -ción almost always have this written accent:

• nación
• información
• organización
• investigación

It also helps distinguish correct pronunciation and spelling.

The basic word order here is:

• Verb + Object
• Terminamos + la investigación

Because the subject nosotros is omitted, the sentence starts with the verb.

If you include the subject, it becomes:

• Nosotros terminamos la investigación.

That is also correct, but less natural unless you want emphasis.

Often yes, but it can also cover research depending on context.

So la investigación might mean:

• the investigation — for example, a police or formal inquiry
• the research — in academic or scientific contexts

For example:

• La policía terminó la investigación. = The police finished the investigation.
• Terminamos la investigación sobre el cáncer. = We finished the research on cancer.

So the exact English word depends on the situation.

A simple pronunciation guide is:

• ter-mee-NA-mos la in-bes-tee-ga-THYON in much of Spain

A few notes:

• z in investigación is not there, but ci before ó gives a th-like sound in most of Spain: -ción ≈ -thyon
• the stress is on:
  • na in terminamos
  • ción in investigación

A rough full pronunciation is:

• tehr-mee-NA-mos la in-bes-tee-ga-THYON

In parts of Spain with seseo or outside Spain, that final part may sound more like -syon instead.