Learn Russian: Оба продукта очень хорошие.

The word оба (both) works just like the number два (two). When used in the nominative case, it requires the following noun to be in the genitive singular. Since продукт is a masculine noun, its genitive singular form is продукта.

If продукта is in the genitive singular, why is the adjective хорошие in the nominative plural?

In this sentence, хорошие is the predicate (it comes after the implied verb 'are'). Even though оба forces the noun into the genitive singular, the subject as a whole ('both products') is conceptually plural. Therefore, the predicate adjective describes a plural subject and takes the nominative plural form хорошие.

What if the noun was feminine instead of masculine? Would оба change?

Yes, the word for 'both' changes depending on the gender of the noun. Оба is used for masculine and neuter nouns. For feminine nouns, you must use обе. For example, if you were talking about cars (машина, a feminine noun), you would say Обе машины очень хорошие.

Could I use the short-form adjective хороши instead of хорошие here?

Yes! Because 'good' acts as the predicate here (stating what the products are), you can definitely use the short form: Оба продукта очень хороши. Short forms often sound a bit more bookish or emphatic in Russian, but both are perfectly correct.