This post is basically a confirmation that Windows 10 installs and runs on the HP Elitebook 8440p. Having said that, on the base configuration (4GB RAM, 5400 rpm hard drive) it does churn the hard disk a lot when you've just logged in, and takes a couple of minutes to boot up. To improve the situation you should increase RAM to the maximum of 8GB and swap out the hard disk for an SSD (solid state disk). When I did this, the boot time reduced to around 20s.
A couple of words of warning with regards the RAM upgrade. Firstly, one of the modules is accessible by lifting out the keyboard; that's not as hard as it sounds, but check the manual. Secondly, make sure that you get the correct speed of RAM. Newer, high speed modules may well not work; again, check the manual for the correct speed for your system.
You could use a disk cloner (found in tools such as EaseUS Todo Backup Free) for switching to the SSD, but if your new SSD is smaller than the old hard disk then you may struggle to free up enough space on the old disk to perform the clone. I would recommend a clean Windows 10 install to the SSD (after a clean Windows 7 install to the SSD, from your original CD/DVD) and then stick your old disk into a 2.5" SATA enclosure (with a USB interface) to copy files across. Do of course make sure that you've got a backup to hand (especially of your irreplaceable personal photos, videos, etc) in case anything goes wrong.
DISCLAIMER: You should only do any of this if you are competent at handling electronic components and administering operating systems. Don't blame me if anything goes wrong when attempting any of this hardware & software work!