The task of establishing the conditions of existence, as well as finding asymptotic images of solutions of differential equations, which contain nonlinearities of various types in the right-hand side, is one of the most important tasks of the qualitative theory of differential equations. In this work, second-order differential equations, which contain in the right part the product of a regularly varying nonlinearity from an unknown function and a rapidly varying nonlinearity from the derivative of an unknown function when the corresponding arguments are directed to zero or infinity, are considered. Necessary and sufficient conditions for the existence of slowly varying $P_{\omega}(Y_0,Y_1,\pm\infty)$ solutions of such equations have been obtained. Asymptotic representations of such solutions and their first-order derivatives have  also been obtained. When additional conditions are imposed on the coefficients of the characteristic equation of the corresponding equivalent system of quasi-linear differential equations, it is established that there is a one-parameter family of such $P_{\omega}(Y_0,Y_1,\pm\infty)$-solutions to the equation. Similar results were obtained earlier when considering second-order equations, which contain on the right-hand side the product of a rapidly varying function from an unknown function and a properly varying function from the derivative of an unknown function when the arguments go to zero or infinity.  Results  for the equation, considered in this paper,  are new.

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