The \textit{Spitzer}/Infrared Spectrograph spectra of three spectroscopically anomalous galaxies (IRAS~F10398+1455, IRAS~F21013-0739 and SDSS~J0808+3948) are modeled in terms of a mixture of warm and cold silicate dust, and warm and cold carbon dust. Their unique infrared (IR) emission spectra are characterized by a steep $\simali$5--8$\mum$ emission continuum, strong emission bands from polycyclic aromatic hydrocarbon (PAH) molecules, and prominent silicate emission. The steep $\simali$5--8$\mum$ emission continuum and strong PAH emission features suggest the dominance of starbursts, while the silicate emission is indicative of significant heating from active galactic nuclei (AGNs). With warm and cold silicate dust of various compositions ("astronomical silicate," amorphous olivine, or amorphous pyroxene) combined with warm and cold carbon dust (amorphous carbon, or graphite), we are able to closely reproduce the observed IR emission of these %spectroscopically anomalous galaxies. We find that the dust temperature is the primary cause in regulating the steep $\sim$5--8$\mum$ continuum and silicate emission, insensitive to the exact silicate or carbon dust mineralogy and grain size $a$ as long as $a\simlt1\mum$. More specifically, the temperature of the $\simali$5--8$\mum$ continuum emitter (which is essentially carbon dust) of these galaxies is $\sim$250--400$\K$, much lower than that of typical quasars which is $\sim$640$\K$. Moreover, it appears that larger dust grains are preferred in quasars. The lower dust temperature and smaller grain sizes inferred for these three galaxies compared with that of quasars could be due to the fact that they may harbor a young/weak AGN which is not maturely developed yet.