We introduce several new results on the Quadratic Eigenvalue Complementarity Problem (QEiCP), focusing on the nonsymmetric case, i.e. without making symmetry assumptions on the matrices defining the problem. First we establish a new sufficient condition for existence of solutions of this problem, which is sonewhat more manageable than previously existing ones. This condition works through the introduction of auxiliary variables, which leads to the reduction of QEiCP to an Eigenvalue Complementarity Problem (EiCP) in higher dimension. Hence, this reduction suggests a new strategy for solving QEiCP, which is also analyzed in ther paper. Next, we present an upper bound for the number of solutions of QEiCP. We also investigate the numerical solution of QEiCP by solving a Variational Inequality Problem (VIP) on the 2n-dimensional simplex, which is equivalent to a 2n-dimensional EiCP. Some numerical experiments with a projection method for solving this VIP are reported, illustrating the value of this methodology in practice.