네가지 is slang for 싸가지, which is also slang. The 싸 in 싸가지 sounds similar enough to the Sino-Korean word for four (사) that it is replaced with the pure Korean word for four (네) to slyly hide its negative meaning, which is ill-mannered or ill-bred. Actually, 싸가지 comes from 싹수, which means good omen or promising, but Koreans almost always say 싸가지 없다, not 싸가지 있다, unless they are joking, so when they just say 싸가지, they mean 싸가지 없다, which is similar in meaning to 버릇없다 or 인정머리 없다.
The word 싹수 can be reduced to 싹, so 싹 can also mean good omen or promising, but 싹 is also the pure Korean word for sprout, so 싹 있다 can also mean, "There are sprouts," something a Korean farmer might say when he sees his crops sprouting in his field, which would be a promising sign or good omen, and this may be the origin of 싹수 있다.
The pure Korean suffix -아지 can be attached to certain nouns to degrade or belittle them, so if Korean farmers see their fields sprouting, they would likely be happy and might say, 싹 있다, "There are sprouts," but if they do not see their fields sprouting, they would likely be unhappy and might belittle their unsprouted sprouts by saying, "싹아지 없다," which is pronounced as /싸가지 없다/ and translated as "There are no damn sprouts."