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We formulate a multi-terminal source coding problem, where we are required to construct a multiple-description code for a source sequence when side information about dependent random processes is available at the decoder only, or at both the decoder and the encoder. We describe an achievable rate-distortion region for these problems in two cases: where there is common side-information at the decoders and when they are different. In the quadratic Gaussian case, and when there is common side information among the decoders, we show that the rate region when both the encoder and decoder have access to the side information coincides with that of decoder-only side information. This is analogous to the single-description (Wyner-Ziv) case, and an explicit characterization of the rate-distortion region is provided for this case.