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Is the image of the general Veronese embedding ever contained in a hyperplane of $P^{n}$? I'm guessing no, but I can't prove it.

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    I'm only familiar with the definition of a sheaf from reading on Wikipedia. I'm working out of Shafarevich's Basic Algebraic Geometry 1, hence I only know about quasi-projective varieties.2011-02-01

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No. To prove it, imagine what it would mean for the image to be contained in a hyperplane: this would mean that some non-zero linear combination of the degree $d$ monomials vanished identically, which is to say, that there is some non-zero degree $d$ homogeneous equation which vanishes identically on $\mathbb P^n$. Hopefully you can convince yourself that this is not possible.