No, it will not damage your computer by deleting the recovery partitions. It will just remove any chance of being able to do a factory restore from them to your original Vista Home Premium that came with the computer. That's all.
I would recommend to create a set of recovery discs as well from the partiton first in case you ever decide you wanted to restore it back on your computer.
If you still want to delete the recovery partitions, then you can use OPTION ONE in the tutorial below to do so. You will need to delete the EASEUS partition (applications & drivers) twice before it becomes unallocated space though.
Partition or Volume - Delete
Afterwards, the two partions will have become unallocated space that you can create a new partition with, or extend the C: partition into if the unallocated space is just to the right of it in Disk Management.
Partition or Volume - Create New
Hope this helps,