Going up in politeness, the phrase yoroshiku onegai shimasu means the same thing, but is used in business settings. It is possible to go further, replacing the polite shimasu with the humble itashimasu, to get yoroshiku onegai itashimasu. In very formal Japanese, such as that used on nengajō (New Year's Greeting Cards), this may be replaced with an even more polite expression yoroshiku onegai mōshiagemasu.
When making requests, at the bottom of the politeness scale comes the plain imperative tabero or kue, literally "Eat!", a simple order to be said to an inferior or someone considered to have no choice, such as a prisoner. This form might convey anger. Similarly, the no da suffix (see What is the no da or n da ending?) can make an order: taberu n da, or kuu n da "Eat!". To express anger, the suffix yagaru also exists: "kuiyagare", an extremely forceful and angry instruction to eat, expressing contempt for the addressee. See What is the yagaru verb ending?
Negatives are formed by adding suffix na: taberu na "do not eat", gomi o suteru na: "do not throw away rubbish". Similarly, the negative of da, ja nai can be used: taberu n ja nai.
More polite, but still strict, is the nasai suffix, which attaches to the i-form of the verb (see What is the 'i' verb ending?). This originates in the polite verb nasaru. Tabenasai thus is an order perhaps given by a parent to a child. This is often colloquially shortened to na, hence tabena.
Requests can also be formed by adding to the te form (see How does the te form work?). The plainest form adds kure, an irregular form of the verb kureru (see What Japanese verbs are irregular?), to the te form. For example tabete kure or kutte kure: "eat it", less forceful than tabero. Negatives are made by negating the te form: tabenaide kure or kuwanaide kure "don't eat it".
Going up one scale in politeness, the more polite verb kudasai is added. For example tabete kudasai. With this polite form, the rough kuu verb is unlikely to be used. Similarly, tabenaide kudasai: "please don't eat it".
A similar entry on the scale of politeness is made by using the imperative form of a polite verb. For example, meshiagaru, the polite verb for "to eat", when turned into meshiagare, the imperative, becomes the response to the set phrase itadakimasu.
Further, more polite forms are also possible. These involve the "i-form" of the verb rather than the "te form", and an honorific prefix. For example, tsukau, "use", becomes o tsukai kudasai: "please use this".
More politeness can also involve indirection of the request: kore o tsukau you ni o negai shimasu "I humbly request that you think about using this".
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