Zernike polynomials are commonly used to describe the aberration of light beams, and the beam quality of Gaussian beams with aberrations can be deduced when the order of Zernike polynomials is limited. In this paper, Hermite polynomials are utilized to reconstruct the aberrations of Gaussian beams. The beam quality factor is directly related to the coefficient and terms of Hermite polynomials and has no limit on its index. We analyzed the beam quality of a Gaussian beam with a quartic aberration and other former 11th Zernike aberrations by Hermite polynomial expansion. The result corresponds with the published research work.