Let \(f(z)\) and \(g(z)\) be transcendental entire functions. W. H. J. Fuchs and G. D. Song proved that if \((f(z), g(z))\) parametrizes some complex algebraic curves, then \(f\) and \(g\) must have a transcendental common right factor. In this paper, we shall prove this result by a different method which also allows us to prove a similar result for some transcendental curves. We then use this result to solve some factorization problems of entire functions.