request vs bespeak

request is a noun but bespeak is not a noun.

Word	Noun	Adjective	Verb	Adverb
request	Yes	No	Yes	No
bespeak	No	No	Yes	No

As verbs, request and bespeak are synonyms defined as:

Other synonyms of request include ask for, call for, quest.

request (noun)	bespeak (noun)
the verbal act of requesting
a formal message requesting something that is submitted to an authority

request (verb)	bespeak (verb)
express the need or desire for	express the need or desire for
ask (a person) to do something	be a signal for or a symptom of
inquire for (information)