I also agree with @Loong, the suggested name "6-ethenyl-4-(prop-2'-enyl) octa-6-en-3-ol" seems incorrect. The correct name for this compound should be (7E)-4-(but-3-en-1-yl)-6-ethenylnon-7-en-3-ol, as @Loong mentioned.
In this compound, the longest carbon chain has 10 carbon atoms, but we select the carbon chain with 9 carbon atoms as it has a functional group (-OH) attached. So, parent chain will be non- as there are 9 carbons.
Now, we number the chain from left to right, as the main functional group is always assigned the least possible locant.
So, we see that we get -OH at locant 3, -CH2CH2CH=CH2 at locant 4, -CH=CH2 at locant 6 and double bond at locant 7. So these substituent/functional groups can be named accordingly:
- (but-3-en-1-yl) (as carbon atom attached to parent chain (alkyl) is always numbered 1)
- ethenyl (vinyl group)
- 7-ene (NOTE: the ending -e will be dropped while writing the complete IUPAC name)
Also, since it is a trans- (w.r.t double bond at locant 7), we put (7E)- in the beginning of the name. (Z is put for cis- )
So, now, the IUPAC name should be (7E)-4-(but-3-en-1-yl)-6-ethenylnon-7-en-3-ol
Here are a few resources:
NCERT Class XI Organic Chemistry - some basic principles and techniques
Systematic IUPAC nomenclature - Aditya Vardhan V
cis- and trans- nomenclature in organic compounds