In this paper, we have suggested a two-step with memory method for solving nonlinear equations by transforming an extant optimal fourth-order without memory method. The acceleration of the order of convergence is attained by employing a single and two self-accelerating parameters. These parameters are estimated by a Hermite interpolating polynomial to enhance the convergence order of {iterative method without memory}. This order of convergence acceleration is achieved without the use of any additional functional evaluations, precisely the convergence order of the suggested two-step with memory method is reached from $4$ to $5.70156.$ The rate of convergence is also verified by Herzberger's matrix method. Finally, various examples are taken into consideration to support the theoretical outcomes.