"look into why" is a correct and usable part of a sentence in written English.
You can use it to ask someone to investigate a particular issue, or to suggest that someone should investigate something. For example, "I think we should look into why our sales have been declining recently."
She said she would look into why company representatives hadn't returned my voice mail message.
As someone who regularly studies people's actions and motivations, did you or anyone look into why that was?
Specifically, the investigators will look into why the median did not prevent the two vehicles from colliding.
It will also look into why relatives were questioned about how much alcohol their loved ones had consumed.
On the analyst call, Mr. Schwartz, in response to a question, said investors needed to look into why rates were rising.
To mark its – nervously awaited – first reissue in Germany since 1945, two Berlin theatre directors decided to look into why Mein Kampf continues to fascinate.
The panel faulted Betsy West for not following up on Heyward's instructions, but did not look into why Heyward did not follow up on his own orders.
"We will have to look into why we did not get expected success," he told reporters in the city of Lucknow.
In his retirement he might want to look into "Why Evolution Is True," being published later this month, which goes over the evidence, some of it brand new.
Alarmed at declining bee populations over the past 15 years or so, particularly in Western Europe, the commission in 2012 allocated resources to 17 member states to look into why bees were in trouble.
Tamil Nadu Chief Minister Jayalalithaa agreed to set up a commission (The Hindu) to look into why the police shot at a group of Dalits on Sunday at Paramakudi in Ramanathapuram, killing six people.
