Hi there
Welcome to the HP Support Forums! It is a great place to find the help you need, from other users, HP experts and other support personnel. I understand that your display is broken on your computer and you are looking for whether it is fixable or if an external display can be hooked up. I am happy to help with this. Please post the full product number for your computer. See the following, if you need help with that information. How Do I Find My Model Number or Product Number?
The quick answer is, yes it is fixable, but it is likely going to need to be serviced by HP, unless you are comfortable with doing it yourself. It is a bit more involved when dealing with an All-in-One computer. As to using an external display, that is also possible, but depends on the model and what ports are available. Some have HDMI ports and others have just USB ports. If there are only USB then it is not as easy to connect an external display when the primary is already unusable.
Anyway, it is likely that you will need to call in to phone support to check on your options. In that case, please use the following http://www.hp.com/contacthp and create a case for your issue and contact HP. If you live outside the US/Canada, please click the link below to get the contact information for your region. http://www8.hp.com/us/en/contact-hp/ww-phone-assist.html
Please let me know how that works for you, and if it does resolve your issue, please mark this post as a solution. Kudos would also be appreciated.