Small-angle neutron scattering (SANS) is a sensitive technique that is able to probe the structure of polymer-grafted nanoparticles and free polymer chains. Here, we combine SANS measurements with self-consistent field theory (SCFT) calculations to study the structure of deuterated poly(methyl methacrylate) (dPMMA) nanocomposites containing PMMA-grafted Fe3O4 nanospheres, with a specific emphasis on the conformation of the PMMA brush chains. We present a new, detailed SANS model which is able to account for the excluded volume of the grafted polymer chains, and compare the experimentally measured brush chain conformations to predictions from SCFT calculations for a polymer-grafted nanosphere, and previous studies by others. The results of the SANS measurements are in qualitative agreement with SCFT calculations and in excellent quantitative agreement with previous studies that indirectly assessed the structure of the polymer chains by measuring the hydrodynamic radius of the nanoparticles. Unlike previous methods, however, SANS provides direct measurements of the polymer brush conformation.