It almost looks like you have an extended partition already. And if you do, you should be able to divide that extended partition up into more partitions (Which they call Logical Partitions) and set up Linux in the Logical Partitions, that's how I installed it.
I had this problem on my laptop, I had all 4 partitions taken up and all of them were being used for something. With a little research though, I found a solution. If you need to make more partitions, you can use an extended partition to hold your other partition. Here's an example (My Computer)
- 220GB NTFS Volume for Windows
- 15GB NTFS Recovery Partition
- 4GB FAT32 Partition for HP Diagnostic Tools
- 199MB SYSTEM Partition
To fix the problem, I got rid of a partition (This was the SYSTEM partition in my case), made changes to my Windows partition to make up for the lost boot partition, booted from the Linux USB (for me it was Kali Linux), and started the installation process.
While installing, it will ask for a partitioning scheme. This is where you select to manually partition it. You should see the unallocated space you made from the deleted partition before (In my case, I had to shrink one of my volumes to make more unallocated space). While selecting to format the unallocated space, you should be able to make an extended partition. When you make it extended, you will see you have 4 partitions, but you can use the free space in the extended partition you made to make more partitions. For me, I had 1 partition left after deleting SYSTEM. I turned the unallocated space into an extended partition and put my swap, root, and home partitions in that extended partition. I now had 7 partitions and Kali Linux installed.