Sum of Tangent Slopes at Triangle Vertices

Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, and $97$. Suppose the curve $V$ with equation $y = (x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $V$ at each of $A$, $B$, $C$.