As J.R. says, that construction is grammatical, and indeed required.
A bare adjective, or one modified by one or more preceding adverbs, goes in front of the noun. (I'm adding a determiner, many, to your sentence, to make the structure a little less ambiguous.)
Many angry people were protesting.
Many passionately and vociferously angry people were protesting.
But an adjective which has a complement cannot be placed in front of the noun; it must be treated as a reduced relative clause† and placed after the noun.
∗ Many angry with the high prices people were protesting.
Many people angry with the high prices were protesting.
If the adjective is modified it will carry its modifiers with it.
Many people passionately and vociferously angry with the high prices were protesting.
The same thing is true of any ‘heavy’ modifier with an embedded complement, such as a participle phrase or an adjectival preposition phrase:
∗ Many suffering from hunger people were protesting.
Many people suffering from hunger were protesting.
∗ Many from the surrounding villages people were protesting.
Many people from the surrounding villages were protesting.
As Laure says, the clause may be bracketed with commas. This changes the meaning, however: the clause is now ‘non-restrictive’ rather ‘restrictive’: being angry with high prices no longer defines the people who were protesting, it is an additional observation about them. A non-restrictive clause can be placed in other places:
Passionately and vociferously angry with the high prices, many people were protesting.
Many people were protesting, passionately and vociferously angry with the high prices.
∗ before an utterance marks it as ungrammatical.
† A reduced relative clause is one from which the relative pronoun and any immediately following copula have been deleted as unnecessary: who were angry with the high prices.
Seek for is not idiomatic English and is a contamination of to look for and to seek. You seek something or you seek to do something, but the preposition for is not normally used in combination with the verb seek. You could say:
- We are seeking people skilled in mathematics.
- We are looking for people skilled in mathematics.
Here is an example of seek + to infinitive:
We seek to expand our collection.
In your second sentence, the adjective different should follow life, because from this one modifies different and not life. Therefore, you should attach it to the adjective instead of splitting the postmodifier into a pre- and postmodifier by putting life in between:
A life different from this one.
There is a difference in usage for pre- and postmodifiers. Premodifiers precede the main noun of your phrase and are normally not too long. Having a long premodifier makes it more difficult to read the phrase. Often, long modifiers are put after the main noun as a postmodifier (although in scientific writing you can still find long premodifiers).
In your example different from this one is a rather long phrase, which is why you should use it as a postmodifier rather than a premodifier. If you only use the adjective different, you can use that as a premodifier. Simple adjectives are often put in the premodifier and longer phrases, such as relative clauses, are used as postmodifiers.
So, in your phrase with skilled, avoid a long premodifier:
We are seeking people skilled in mathematics.
In any case, do not split your adjective from other phrases that are modifying it:
We are seeking skilled people in mathematics.
In the above sentence, skilled still modifies people, but in mathematics no longer modifies skilled. As a result, you're saying that in mathematics is the location where you will be looking for skilled people.
Both of your sentences with long premodifiers sound odd. If you have no postmodifier, don't put in a long premodifier. You could perhaps make your premodifier slightly longer if your postmodifier is already terribly long, but in general the premodifier restricts itself often to just some adjectives, except for scientific writing as I mentioned earlier:
- We seek a much more pleasant life different from this one.
- We seek a much more pleasant life that is totally different from this one.
- We seek extraordinarily qualified people skilled in mathematics.
- We seek extraordinarily qualified people who are skilled in mathematics.
Note how the premodifier more or less restricts itself to adjectives (and adverbs) while relative clauses, gerunds and other phrases often go in the postmodifier. In all four of the sentences above, the premodifier could be moved and used as a postmodifier whereas it would not really work the other way around.
Additionally, it sounds rather odd to mention from this one before you mention life.
In short, don't make your premodifier too long, it will not improve readability. Restrict it to mostly adjectives and perhaps adverbials modifying those adjectives. Longer phrases should go in the postmodifier.
Best Answer
The problem is that grammar is somewhat tied to meaning here. The position of an adjective in a sentence depends on its role.
When used attributively (to describe a noun), as stated in other comments and answers, the adjective comes before the noun:
If you say:
This can become:
At first glance this doesn't really seem to change the meaning since:
rivers that are navigable = navigable rivers
Edit: But...
When an adjective comes after the noun it describes (like in the 3rd example), it functions as a postpositive modifier. Changing the position of the adjective (relative to the noun it describes) may bring a slight difference in the meaning of the sentence (the meaning of the word itself does not change!). When used postpositively an adjective connotes an ephemeral quality, one that is present at the moment, but doesn't always have to be. On the other hand, the adjectives used attributively may express either an ephemeral or a permanent characteristic, depending on the context. The difference between attributive and postpositive use of an adjective is explained in more detail in (the middle of) this post and in the comments.
Only some adjectives can be used both attributively and postpositively (while retaining the same word meaning), and these are the ones ending in -able and -ible (such as navigable). (But not even all of those - see later: responsible).
To cover another aspect (this is where grammar kicks in again): if an adjective is used predicatively (in a pattern: subject + verb + object + complement (here an adjective)) it would be in a sentence like this:
The meaning of some adjectives (when used as modifiers) changes depending on whether they are used attributively or postpositively. Some examples are: concerned, responsible, present etc. Neither navigable nor excited are among those. Here the meaning of the word itself changes and the difference can be determined by checking the dictionary definitions.