The context here is "You should record your speech". I mean you should record both your audio (the speaking) as well as the visual (gesticulation, etc.), but the context of this sentence strongly suggests that only audio is intended. This sentence appears in sales copy and so word count is limited. "You should video record your speech" sounds clumsy. How can I phrase this smoothly and succinctly?
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You should film your speech. Audio recording is understood as included.
While I agree with Jon Hanna's comment that you could use video as a verb, it might be less clumsy to say
We used to use videotape as a verb, but technology has pretty much rendered that an anachronism.
While film still exists, it is rarely used in amateur recording. As a verb it is understandable, even if inaccurate.