As the title suggests, this lesson will be about compound verb endings for fixing or making errors of judgment.
忘れる, which is an 一段 verb, means "to forget". In compound verbs it means "to forget to...". This implies that you forgot to do something that you needed to do. If this is not what you wish to imply, ～のを忘れる should be used. Both are often used with ～てしまう, which means "to accidentally/end up doing".
I forgot to turn off the TV.
I forgot to buy the cake.
I left the company without turning off the printer.
4b. 宿題（を）するのを忘れた。 (もっと自然)
I accidentally forgot to do my homework.
Naturalness Note: The notes on naturalness are most relevant when there is no preceding context. If you were to tell your Japanese teacher, for instance, that you forgot to do your homework, it would be more appropriate to use 4a.
Isn't it embarrassing to forget your lyrics?
I completely forgot.
I'd completely forgotten to call him.
I forgot to take a picture of the waterfall.
直す is used in a variety of situations to represent that one is curing or fixing something. In compound verbs it expresses that one is "doing something over again".
To call back.
I planned it over again.
I redid it.
Orthography Note: ～直す is often left in かな.
間違える is the transitive form of 間違う and means "to confuse/mistake". The intransitive 間違う is sometimes used nowadays to function as 間違える, but when this happens, it tends to be in very casual speech. In compound verbs, ～間違える is used to mean "messed up on...".
I made a mistake in writing my name.
I made a mistake in speaking.
I mistakenly sang the wrong song.
I messed up on the song.
Phrase Note: Another similar phrase would be 歌で失敗した meaning "failed at the song".
He mistook me for his little brother.