42. \( 3 x^{2}+y^{2}+18 x-2 y-8=0 \)

To deepen your understanding of this equation, it represents a conic section, specifically a hyperbola or an ellipse, depending on the coefficients when rearranged. In this case, on simplifying and completing the square for both \(x\) and \(y\), you can identify the shape and orientation of this conic. This will provide a graphical insight into how these equations manifest in the coordinate plane! When tackling equations like this, a common mistake is to overlook the need for rearranging terms or not completing the square correctly. Make sure to carefully isolate terms and rearrange them, paying attention to the signs. Often, the form can seem daunting at first, but cleanly organizing your equation can clarify whether you're dealing with an ellipse, hyperbola, or some other conic!