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An efficient technique for the analysis of electromagnetic scattering by arbitrary shaped inhomogeneous dielectric objects is presented. The technique is based on a higher-order method of moments (MoM) solution of the volume integral equation. This higher-order MoM solution comprises recently developed higher-order hierarchical Legendre basis functions for expansion of the electric flux density and higher-order geometry modeling. An unstructured mesh composed by trilinear (8-node) and/or curved (27-node) hexahedral elements is used to represent the dielectric object accurately. It is shown that the condition number of the resulting MoM matrix is reduced by several orders of magnitude in comparison to existing higher-order hierarchical basis functions and, consequently, an iterative solver can be applied even for high expansion orders. Numerical results demonstrate excellent agreement with the analytical solutions for the dielectric sphere, as well as with results obtained by other numerical methods.