Do you not think it has made our lives better?

Don't you think it has made our lived better? It's incorrect to say "Do not you...?"

These sentences are negative questions.  The former is uncontracted, which is formed in the construct of "auxiliary + subject + not", whereas the latter is contracted, which is formed in the pattern of "auxiliary + 'nt + subject". Both the types are grammatically correct. The only difference is that the former type is formal. On the other hand, the latter is more usual and used in formal and informal English.