How can we avoid ___ mistakes in English?

The correct choice is "embarrassing," which is the proper form of the word to use in this context. This adjective describes mistakes that are likely to cause embarrassment. The spelling of "embarrassing" includes a double "r" and a double "s," which is the standard and widely accepted form in English.

The other options present variations that are incorrect or do not fit the context. "Embarrasing" and "embarasing" are both misspellings of the word. "Embarrassing" must have two "r"s and two "s"s to be correct. The option "embarrased" is also incorrect because it is the past participle form of the verb "embarrass," and it does not serve as an adjective to describe the type of mistakes one might want to avoid. Therefore, the choice of "embarrassing" is not only spelled correctly but is also the appropriate adjective for the type of mistakes being referenced.