In addition to amazing explanation by k_tron_3000, I'd like to add that patient also has numbness and tingling in both legs. So, all sensations to the spinal cord are impaired, hinting towards damage to dorsal horns and hence sensory neuropathy.
Romberg sign + = Sensory ataxia
submitted by โk_tron_3000(35)
The description of bilateral lower limb loss of vibration implies DCML damage, and the absent DTRs + Romberg seem to me to be implying that he possibly has tabes dorsalis from syphilis (or something very similar in presentation).
As for the other answers, A is wrong because his motor function is intact, B is wrong because pain and temperature deficits are not mentioned, C is wrong because it implies a specific nerve is entrapped, but he has lost bilateral sensation in his entire lower extremities
D is the trickiest, and Iโm not 100% sure, but I would think radiculopathy of the anterior (ventral) roots would cause motor deficits since they carry motor efferents. You might also expect that motor dysfunction to be unilateral, since it would be unlikely to have a problem with the nerve roots on both sides. also the DCML is not located near the anterior roots of the spinal cord, so if the anterior roots were affected you really wouldnโt expect to see vibratory loss.
So basically process of elimination, I do feel like sensory neuropathy is an extremely vague answer though and I wasnโt a fan of the question.