An extended non-linear Schrödinger equation (NLSE) combining quadratic and cubic Non-linearities, which appears as an approximate model of a relatively dense quasi-one-dimensional Bose–Einstein condensate (BEC), is considered. In particular, we focus on the most physically important situation where the external potential and the quadratic-cubic non-linearities are dependent on both time and spatial coordinates. We use the similarity transformation technique to construct novel exact solutions for such NLSEs with modulating coefficients. We first present the general theory related to the quadratic-cubic model and then apply it to calculate explicitly soliton solutions of W-shaped, bright and kink type. The dynamic behaviors of solitons in different non-linearities and potentials that are of particular interest in applications to BECs are analysed.