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For nonnegative real numbers $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$such that $B+C>0$ and $\alpha+\beta+\gamma >0$, the difference equation  \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gammax_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,... %, \quad x_{-1},x_{0}\in[0,\infty) \end{equation*} has a unique positive equilibrium. A proof is givenhere for the following statements: \medskip  \noindent Theorem 1. {\it For every choice of positive parameters $\alpha$,$\beta$, $\gamma$, $A$, $B$ and $C$, all solutions to the difference equation  \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gammax_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in[0,\infty) \end{equation*} converge to the positive equilibrium or to a primeperiod-two solution.} \medskip  \noindent Theorem 2. {\it For every choice of positive parameters $\alpha$,$\beta$, $\gamma$, $A$, $B$ and $C$, all solutions to the difference equation\begin{equation*} x_{n+1}= \displaystyle\frac{\alpha +\beta x_{n}+\gammax_{n-1}}{B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in(0,\infty) \end{equation*} converge to the positive equilibrium or to a primeperiod-two solution.}

Global Behavior of Solutions to Two Classes of Second-Order Rational Difference Equations