Tangent to Cubic: Open Middle Problem

Creation of this [url=https://www.openmiddle.com/]Open Middle[/url] problem was inspired by [url=https://twitter.com/Cshearer41]Catriona Agg[/url].
In any app below, enter coefficients and move the point around to make the statement in this picture true. How many solutions can you find?
Can you find another solution different from the one above? Try to do so.
Can you find another solution different from the other two above? Try to do so.
– Tim Brzezinski

Information: Tangent to Cubic: Open Middle Problem