Suppose I am considering buying a used car:
"We carefully checked the car for defects."
"We carefully checked out the car for defects."
What is the difference between "check" and "check out"?
With "check out" you don't really need to elaborate.
We checked the car for defects = We looked for defects (if any) in the car.
We checked out the car = We carefully inspected the car (the unspoken suggestion is that you would be checking there was nothing wrong with it.)