Consider the curve defined by $2y^3 + 6x^2y - 12x^2 + 6y = 1$.
(a) Show that $\dfrac{dy}{dx} = \dfrac{4x - 2xy}{x^2 + y^2 + 1}$.
(b) Write an equation of each horizontal tangent line to the curve.
(c) The line through the origin with slope $-1$ is tangent to the curve at point $P$. Find the $x$- and $y$-coordinates of point $P$.