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We present a family of four-point quadrature rule, a generalization of Gauss-two point, Simpson's 3/8, and Lobatto four-point quadrature rule for twice-differentiable mapping. Moreover, it is shown that the corresponding optimal quadrature formula presents better estimate in the context of four-point quadrature formulae of closed type. A unified treatment of error inequalities for different classes of function is also given.