5) \( x^{2}+8 x y+15 y^{2} \)

This expression represents a quadratic in two variables, \( x \) and \( y \). One fascinating takeaway is that it can be viewed geometrically as a conic section, specifically an ellipse in certain conditions. To analyze these types of expressions, one often completes the square or uses techniques such as substitution to identify its shape and properties, further illuminating the relationship between the variables. When working with expressions like \( x^{2}+8 x y+15 y^{2} \), it’s vital to check for common factors and look for potential mistakes. A classic pitfall is neglecting to fully factor or misapplying the quadratic formula for multi-variable equations. Instead of panicking over complexity, break it down into manageable parts, and remember, practice is key!