I learned that after words like "einige, mehrere, etc." the following adjectives also have the same declination. However, I got corrected when I said "mit einigen meinen Freunde" to "einigen meiner Freunde". Why is it "einigen meiner" and not "einigen meinen"?
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Maybe it helps to look at the English-language equivalent:
As you may know, mit (English: with) inflects what follows it to the dative case (einige --> einigen). And of can be expressed in German with its counterpart von, for example:
As with mit, von inflects the following noun phrase to the dative case. Concatenating mit and von like so
is grammatical German but considered inferior style, especially in writing. (Some people may disagree and consider the double dative acceptable.)
Instead, you would drop the preposition and inflect the noun and its possessive pronoun to the genitive case, i.e., meiner Freunde (and not meinen Freunden, which is dative case). This rule supersedes the otherwise correct rule that you learned about matching the declination.
UPDATE: I am reliably informed that I erred in the "bonus" part of this Answer.
In response to my own question, "But what if you stick in an adjective?"
I had offered
That was correct. However, I erred in identifying "guten" as dative plural. It is in fact genitive plural, the same genitive plural as "meiner" and "Freunde". My mistake shows how tricky identifying the case of individual words can be when different cases can be letter-for-letter identical and only functional analysis will differentiate them.
What happens in your example is that meine turns Freunde from an indefinite group (some friends) to a definite group (all my friends). The same thing would happen if you used diese, jene etc. instead of meine. Applying einige (or mehrere) to a definite group makes no sense, therefore meine Freunde is put into the genetive case and the whole term einige meiner Freunde means (with maximum explicitness) some members of the group which consists of all my friends or just some of my friends.
The same happens in English, where you can say some friends or some of my friends but not some my friends. I could not find an example on the fly where German does not behave analogously to English concerning this.