The authors solve a fourth-order equation with the spectral Galerkin method. Their discussion is restricted to the two-dimensional case where this gives rise to a linear system of equations with the following structure: \[ M u^{h} \equiv (A \otimes B +2C \otimes C +B \otimes A + \alpha (C \otimes B + B \otimes C) + \beta B \otimes B)u^{h} =f^{h} \tag{1} \] An \(O (N^{3})\) method for solving (1) is presented, and it is shown that in the case of many right-hand sides the solution of (1) with a Chebyshev basis is preferable to the solution with a Legendre basis.