What complex numbers satisfy $\lvert z-1 \rvert \lvert z+1 \rvert< 3$?
The set of complex numbers satisfying $\lvert z-1 \rvert=r$ for $r$ a positive real number is the open disc centered at $1$ with radius $r$. How to proceed? Is this set connected?
This kind of curve where the product of distances to two fixed points is constant is called a Cassini oval. It is not an ellipse, though looks like one if the constant is at least $2$.
Since you wrote an inequality, the set of satisfying points is everything strictly within the oval.