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The phrase "make a formal complaint" is correct and commonly used in written English.
It can be used when a person wants to formally express their dissatisfaction or grievance with a certain situation, person, or organization. This phrase is often used in professional settings, such as in the workplace or when dealing with customer service. Example: "I would like to make a formal complaint about the subpar service I received at your restaurant last night. The food was cold and the waitstaff was extremely rude."
Exact(60)
"We're going to have to make a formal complaint.
Barcelona has decided not to make a formal complaint about the assault on Vilanova.
The woman consulted staff counsel and colleagues for months before deciding to make a formal complaint.
The problem is that no one is prepared to make a formal complaint.
If you say anything they tell you sarcastically to go make a formal complaint.
However, in practice no civil servant is likely to make a formal complaint against a minister.
The NUT planned to make a formal complaint to Ofsted, she added.
He is also said to have threatened to write to the Metropolitan police commissioner to make a formal complaint.
She had previously taken the matter to Albany detectives, who said she did not make a formal complaint.
If the retailer refuses to give you a refund, repair or replace your item, you should make a formal complaint.
A child's right to make a formal complaint is now embedded in social care policy and practice.
More suggestions(15)
make a formal commitment
make a formal occasion
make a formal presentation
make a formal bid
make a formal nomination
make a formal approach
make a formal reply
make a formal accusation
make a formal asylum
make a formal appeal
make a formal judgment
make a formal offer
make a formal apology
make a fucking complaint
make a formal proposition
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com