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The Green-Griffiths-Lang conjecture predicts that varieties of (log) general type are hyperbolic outside of a proper subvariety called an exceptional locus. We prove an algebraic version of this Conjecture, with respect to Demailly\u2019s algebraic version of hyperbolicity, for the complement of a very general degree 2n hypersurface in Pn. Moreover, for the complement of a very general quartic plane curve, we completely characterize the exceptional locus as the union of the flex and bitangent lines. 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