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Since the discovery of the cosmic X-ray background (CXB), astronomers have strived to understand the accreting supermassive black holes (SMBHs) contributing to its peak in the 10-40 keV band. Existing soft X-ray telescopes could study this population up to only 10 keV, and, while NuSTAR (focusing on 3-24 keV) made great progress, it also left significant uncertainties in characterizing the hard X-ray population, crucial for calibrating current population synthesis models. This paper presents an in-depth analysis of simulations of two extragalactic surveys (deep and wide) with the High-Energy X-ray Probe (HEX-P), each observed for 2 Ms. Applying established source detection techniques, we show that HEX-P surveys will reach a flux of ~10-15 erg s-1 cm-2 in the 10-40 keV band, an order of magnitude fainter than current NuSTAR surveys. With the large sample of new hard X-ray detected sources (~2000), we showcase HEX-Ps ability to resolve more than 80% of the CXB up to 40 keV into individual sources. The expected precision of HEX-Ps resolved background measurement will allow us to distinguish between population synthesis models of SMBH growth. HEX-P will leverage accurate broadband (0.5-40 keV) spectral analysis and the combination of soft and hard X-ray colors to provide obscuration constraints even for the fainter sources, with the overall objective of measuring the Compton-thick fraction. With unprecedented sensitivity in the 10-40 keV band, HEX-P will explore the hard X-ray emission from AGN to flux limits never reached before, thus expanding the parameter space for serendipitous discoveries. Consequently, it is plausible that new models will be needed to capture the population HEX-P will unveil. [ABSTRACT FROM AUTHOR]