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"?
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.)