Sorry, that was the short answer - my wife was calling to say that my tea was out on the table!
It depends upon how important the machine is to you.
For example, is it an important machine (e.g. a prototype computer). In this case probably not. An HP-85 was a consumer product...
Was it owned by (for example) a family member. I suspect not from your initial post.
Did it cost you much money, or was it given to you free?
If it goes 'bang' when you do power it up - is that of any consequence to you?
Only you can answer the above.
Most importantly is your safety. As this will be plugged into the mains power two things could happen: (1) electrocution and (2) fire.
I would test the machine for electrical safety first using a PAT tester (sic). This ensures that the mains electrical insulation is in good order and that the protective earth (if fitted) is in good order.
I would then take the machine apart and look at the condition of the RIFA capacitors, electrolytic capacitors and tantalum bead capacitors. of course, clean it out with a vacuum cleaner and get rid of the dead bugs and dust as well at the same time! I would then check for signs of damage or burning / heat.
However, other people (myself included) just power the things up and hope!
I had a load of equipment to dispose of a while ago. I just took the equipment outside (on a nice sunny day), plugged it into a disconnected extension lead and turned the equipment on. I then went inside and plugged the extension lead into the mains - being a suitable distance away! If the thing went BANG or burst into flames - it got skipped. If it did nothing - it was given away as none working. If it seemed to work, it was given away as may work...
For important machines (however) I don NOT recommend this course of action!
The main culprits are capacitors.
RIFA capacitors are on the mains side and crack. They let out horrid-smelling smoke, can go BANG and can also burst into flames.
Electrolytic capacitors can leak and dry up. If they start to break down and start to 'gas' they can also go BANG.
There is very little easy way of testing tantalum bead capacitors. They are known as "smurf grenades" and can also go BANG as well.
If you fancy just powering it up, make sure the mains voltage is correct for the machine and your mains supply and ensure it is in a well ventilated area and you are not in the immediate vicinity...
Dave