The author considers a first order differential equation of the form \[ f'=P_0+P_1\cdot f+\dots+P_n\cdot f^n,\quad n\geq3 \tag{1} \] where each \(P_k\) is a polynomial, \(P_n\neq0\). Since \(n\geq3\), it follows from the Malmquist Theorem that any meromorphic solution of equation (1) must necessarily be a rational function. Furthermore, it was shown by G. G. Gundersen and I. Laine that (1) can possess at most a finite number of distinct meromorphic (rational) solutions. In the paper new estimates for the maximum number \(M\) of distinct meromorphic solutions and also for the maximum number \(L\) of linearly independent meromorphic solutions of the equation (1) are found. The estimate for \(M\) depends only on \(n\) and the number \(d\) of distinct zeros of \(P_n\), while the estimate for \(L\) depends only on \(d\).