Despite being a major pillar of galaxy evolution, galactic feedback from stars and supermassive black holes (SMBHs) is subject to very little observational constraint. This is particularly true of the hot component, as viewed in X-rays. Yet, the hot component is directly linked to much of the energetic feedback released from these compact objects. X-ray observations suffer from several challenges that make placing this constraint a difficult task. In the face of considerable model uncertainty, these challenges underscore the need for novel X-ray data analysis techniques. In this dissertation, I seek to lend a unique perspective to X-ray data analysis and initiate the steps towards unravelling the hot component of galactic feedback. This is done through spatio-spectral fitting with Markov Chain Monte Carlo (MCMC). First, I fit 2D simulations of SMBH accretion to three separate bands of Chandra imaging data of Sgr A*, the SMBH at our galactic center. In this study I place the first observational constraint on the angular momentum of accreting gas and self-consistently deconvolve residual point-like emission from the spatially extended accretion flow. I extend this analysis in Appendix 2 by re-examining the spectral energy distribution of Sgr A* from radio to X-ray. I find that a 1D accretion flow model cannot be reconciled with the more detailed X-ray modelling results. I further speculate on the origin of very steep synchrotron emission, suggesting that the residual point-like emission is accelerated by magnetic turbulence. Second, I describe the methodology for extracting spatial information from the RGS grating spectrometer onboard the XMM-Newton satellite. I demonstrate this method using 32 observations of M31 by fitting the OVIII Ly-alpha and OVII K-alpha transitions. I show that the observed spectral peculiarities are much more likely the result of resonance scattering, rather than SMBH feedback effects seen through plasma overionization. A semiparametric extention of that work is also provided in an appendix. Finally, I conclude with a discussion of the usefulness of spatio-spectral analysis and highlight the promising research toward understanding galactic feedback that can be done as an extention to the work herein.