Chemistry Questions...?

The stronger base among the 4 given compounds is (H₃C)₂NH.

The 4 given compounds are amines which are basic because the unshared pair of electrons on the N atom can accept a proton readily.
Amine + H⁺ ⇌ Ammonium ion

We firstly rule out H₅C₆NH₂ which is the weakest base. This is because the unshared electron pair on the N atom participates in the delocalization of  electrons on the benzene ring. Thus, the unshared electron pair on the N atom is less available to accept a proton. Besides, the delocalization energy is decreased when the N atom accepts a proton.

For the rest 3 aliphatic amines, their basic strengths are governed by two counter factors.

(1) Alkyl groups are electron-releasing which would increase the stability of the ammonium ion by inductive effect (making the positive charge more dispersed), and thus the presence of alkyl group(s) would increase the basic strength of aliphatic amine.
Due to this effect, basic strengths : NH₃ < H₃CNH₂ < (H₃C)₂NH

(2) Alkyl groups would decrease the stability of the ammonium ions by lowering their hydration enthalpies, and thus the presence of alkyl group(s) would decrease the basic strength of aliphatic amine.
Due to this effect, basic strengths : NH₃ > H₃CNH₂ > (H₃C)₂NH

Due to the above two counter effects, basic strengths : (H₃C)₂NH > H₃CNH₂ > NH₃

Therefore, (H₃C)₂NH is the strongest base among the 4 given compounds.

The easiest way to answer this is to look at a table of Kb values for these compounds. The strongest base will have the largest value of Kb. In my text, that would be dimethylamine.