Nanosized particles with high responsivity in the infrared spectrum are of great interest for biomedical applications. We derive a closed-form expression for the polarizability of nanoparticles made of up to three concentric nanolayers consisting of a frequency dependent polar dielectric core, low permittivity dielectric spacer shell and conductive graphene outer shell, using the electrostatic Mie theory in combination with conductive layer in a dipole approximation. We use the obtained formula to investigate SiC, GaN and hBN as core materials, and graphene as conductive shell, separated by a low-permittivity dielectric spacer. Three-layer nanoparticles demonstrate up to a 12-fold increased mid-infrared (MIR) absorption as compared to their monolithic polar dielectrics, and up to 1.7 as compared to two-layer (no spacer) counterparts. They also show orders of magnitude enhancement of the nanoparticle scattering efficiency. The enhancement originates from the phonon-plasmon hybridization thanks to the graphene and polar dielectric combination, assisted by coupling via the low permittivity spacer, resulting in the splitting of the dielectric resonance into two modes. Those modes extend beyond the dielectric's Reststrahlen band and can be tuned by tailoring the nanoparticles characteristics as they can be easily calculated through the closed-form expression. Nanoparticles with dual band resonances and enhanced absorption and scattering efficiencies in the MIR are of high technological interest for biomedical applications, such as surface -enhanced vibrational spectroscopies allowing simultaneous imaging and spectroscopy of samples, as well as assisting guided drug delivery.