From your description and fdisk -l data, you have a free space in between /dev/sda2 and /dev/sda3.
unfortunatedly this arrangement is not going to work for adding Linux OS, as you already found out.
What you can do is to use gparted or other partition program to 'shift' the partitions so that /dev/sd3 will become adjacent to /dev/sda2.
Then to overcome the limit of 4 primary partitions you must delete the /dev/sda4.
On HP_TOOL partition, I do not know what are these, but from the wording HP_TOOL, it might just be some softwares since it is just 103Mbytes, and this is the one that was 'designed'
to prevent you to have ease of Linux installation.
If I were in the position, I would copy these files outside the computer for back up and at the same time just copy into the C: drive.
Then use gparted to delete this /dev/sda4. Once you do that you should see a large unallocated space on the right. This is when you become '
free from that 4 limit'.
Now you instruct gparted to change
ALL the unallocated space to a
Extended partition, it will assign /dev/sda4 to it, note the difference is this is
Extended partition (not Primary) and it will take any number of logical partitions ( as many as you can put in these space, of course there is a limit but that is about 3 digits for new kernel and some old kernel has 16).
Next is just add new logical partition such as
1. a Linux Swap of something like 2x your RAM size
2. another bigger partition says 10G for one Linux OS, or 20G for more space
3. add more if you need multi boot.
and enjoy Linux installation.
good luck
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