smc

Graph the two equations. $y=(x+1{)}^2$ produces a parabola with vertex $(-1,0),$ while $xy+y=1$ produces a hyperbola. They intersect at the point $(0,1).$ This makes one real solution. Only one answer choice has 1 real solution listed, so the answer is $\boxed{\text{1 real and 2 imaginary pairs}}.$