It would be more common to say "please, check this for me". "check it out" is more commonly used to direct someone's attention to something. For example, you just bought a new phone and show it to a friend, saying "check it out". "check it out" does not typically mean "verify" as "check" does.

You would normally say "can you look into this for me" or "can you look into this?" People also say "check this out" but it usually means going to see something for the first time, not a verification of a previous notion (a double check). I am going to check out the new store around the corner, I hear they have great coffee. They say the print shop down the street is cheaper, so I am going to check this out on my break and we can decide on whether we should print there or not, after lunch. If you read something and you want to make sure it is true, you can ask someone to "double check" it. Can you double check this? A Cross check is an illegal check in ice hockey. I know it can also mean to verify by a different method, but in Canada, the first thing one might think of is this: https://www.youtube.com/watch?v=LKIKf7PXjnQ . So, one is for initial viewing, double check is for re-confirming, and cross check is to reinforce with an additional method of verification. So, I can go online to check out the prices, double check by having Steve go online, and cross check by calling the store and asking what their prices are.

Igor mades a useful point: "check it over" can be used in the same way to mean "verify". "please check this for me" and "please check this over" mean the same and both are used commonly. Thanks Igor. -S