I have understood how this is acceptable in USA and in many parts of the world. This concept of plea bargains and not having to prove anything beyond a certain

I have understood how this is acceptable in USA and in many parts of the world. This concept of plea bargains and not having to prove anything beyond a certain limit. I mean isn't it clearly the notion that miscarriage of justice is accepted? I accept that a lot of this idea in my head is from films set in the USA but then again I have read on the subject and plea deals seem to be not only accepted but encouraged. How a person, who seemed to have murdered someone, ought to be given only M years of sentence just because that person agrees for a plea deal where if properly tried and proven guilty the person might have been sentenced for N years which usually would be more than M? Just because the prosecution have to work less the culprit gets a lesser sentence? And as you say many who are arrested and haven't done anything but just wants to live with a lesser sentence just agree to such deals because they can't afford a lawyer, let alone a good lawyer. Is it not questioned? Is it something in the constitution? How does it work? The "deal" itself - is it verbal or plea deals are actually signed and verified?