Two new hard-sphere equations are proposed which, in combination with a van der Waals attraction term, lead to a biquadratic, respectively a cubic, equation of state. The new equations show the correct limiting behavior at low as well as at high densities; their poles are close to the physical packing fraction of hard spheres. Both equations of state were extended towards mixtures by one-fluid mixing rules, and their global phase behavior was investigated for the special case of equal-sized molecules. Both equations are able to predict closed-loop liquid-liquid immiscibility; the topology of the phenomenenon is the same as for the Carnahan-Starling equation. It appears the occurrence of closed-loop liquid-liquid immiscibility does not depend on the location of the pole nor on the degree of the equation of state used.