骨が折れました

Toritoribe said:

犬を見つけるのは骨が折れたので、お礼に猫を貰いました。

I think 犬を見つけるのは難しかったので、お礼に猫を貰いました。 would be more common.

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The 「の」 particle in this usage essentially replaces the noun and takes over the role as a noun itself. We can essentially treat adjectives and verbs just like nouns by adding the 「の」 particle to it. The particle then becomes a generic noun, which we can treat just like a regular noun.
（１） 白いのは、かわいい。- Thing that is white is cute.
（２） 授業に行くのを忘れた。- Forgot the event of going to class.

Now we can use the direct object, topic, and identifier particle with verbs and adjectives. We don't necessarily have to use the 「の」 particle here. We can use the noun 「物」, which is a generic object or 「こと」 for a generic event. For example, we can also say:
（１） 白い物は、かわいい。- Thing that is white is cute.
（２） 授業に行くことを忘れた。- Forgot the thing of going to class.

However, the 「の」 particle is very useful in that you don't have to specify a particular noun. In the next examples, the 「の」 particle is not replacing any particular noun, it just allows us to modify verb and adjective clauses like noun clauses. The subordinate clauses are highlighted.
（１） 毎日勉強するのは大変。 - The thing of studying every day is tough.
（２） 毎日同じ物を食べるのは、面白くない。- It's not interesting to eat same thing every day.

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Toritoribe said:

That の is not a particle but a nominalizer. の turns 犬を見つける into a noun clause, just like "I search a dog" --> "serching a dog/to search a dog" in English.

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Toritoribe said:

Do you mean something like "I wanted to have a dog. I looked for it but it's hard to find. Then, I changed my mind and decided to have a cat" by your sentence? If so, I misinterpreted the meaning because of the nuance of the words ﾅ陳ｩ窶堙や?堋ｯ窶堙ｩ and 窶禿｡窶堋､.
My sentence is for "You looked for someone's missing dog, but it's hard to find. After searching, he/she gave you a cat in return for your help".

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Toritoribe said:

母は犬を見つけられなかったので、代わりに、私に猫をくれました。

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Toritoribe said:

Since my mom couldn't find any dogs for me, she gave me a cat instead.
母は犬を見つけられなかったので、代わりに、私に猫をくれました。


If the clause is 犬を見つけられなかったので, i.e., when you don't state who the subject is, the subject of 見つけられなかった is interpreted as the speaker "I" or "general you".

It was so hard to find a dog that I got a cat.
犬を見つけるのは難しかったので、代わりに猫を貰いました。
("Generally speaking, it was hard to find a dog" or "It was hard for me to find a dog", therefore I got a cat from someone instead.)

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Rei Yahya said:

I'm so sorry to interject, but this conjugation, 'mitsukerareru'...~rareru and ~reru I've heard explained two ways, one is that it's just another version of Base 4, and another that it's used when you want to give a certain special respect to whoever you're talking about. Or is it both of those combined? :? Thank you!

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Toritoribe said:

Yes. It's just the difference between 難しい and 大変（だ）, ので and から.The structure is the same.

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