DId you try another slot and see if the AMD HD 7850 still have same issue? DId you try another video card on PCI 16 slot to see if it works? I want to make sure it is not faulty video card.
Asus motherboards have a 3 year warranty but you need to check on your product page to see if there are any exclusions for it.
You have to register your product with ASUS tying it to your account and then apply for rma through them.
I went through the process earlier this year with my t100 transformer tablet and it was pretty easy.
They keep you updated on the progress of the repair and then send it back when they are done.