(on mobile, so sorry for any formatting weirdness)

English teachers will only give you an arbitrary, subjective answer about whether it’s a word - you want a linguist if you want an objective answer.

Since we’re dealing with two different “words” (roots) here, factory and overclocked, the first thing to look for is compound stress. Many compound words in English get initial stress: compare “blackbird” and “a black bird”.

This isn’t foolproof, however. For some speakers there are compounds that don’t get compound stress - some speakers say “paper towel” as expected, while others say “paper towel”, but it’s still a compound either way.

So how can we actually tell that paper towel is one word? See if the first member of the potential compound (the non-head) can be modified in any way.

For example, we know doghouse is a compound because in “a big doghouse” big can only refer to the house, and cannot refer to “the house of a big dog”. Similarly, blackboard must be one word because it can take what appear to be contradictory modifiers: " a green blackboard".

So, in the same way, paper towel and toilet paper are one word because “big paper towel” can’t mean “a towel made from big paper” and “pink toilet paper” can’t mean “paper for a pink toilet”. (Toilet paper also gets compound stress.)

Yet another way to test is by semantic drift (meaning shift). As mentioned earlier, blackboards don’t have to be black, so the meaning of the compound doesn’t perfectly correspond to the pieces of the word - instead, the fact that it’s a vertical board you write on in chalk is much more important to the meaning. This is because once the pieces combine to form a new word, that new word can start to shift away from the meaning of the pieces. Again, however this process takes time, so it’s not a perfect test.

So, back to the original question: is “factory-overclocked” one word?

Well, it doesn’t get compound stress, and for me I can still say things like “it’s home-factory-overclocked” to mean that it was overclocked in its home factory, so the first member can take modifiers. And, the whole thing still means what the pieces mean.

So, in my grammar, “factory-overclocked” is two words. But for some of you “home factory overclocked” may not be possible, which would indicate that it’s started to become one word for you. Everyone’s grammar is different, but we can still test for these categories.

If you instead mean by your question, “can factory and overclocked be combined with a hyphen?”, however, I can’t help you, because language-specific writing conventions are subjective and arbitrary, and not something that linguists usually care very much about.