\( \left. \begin{array} { l } { ( x + 4 ) ( x - 4 ) } \\ { ( 3 y + 2 ) ( 3 y - 2 ) } \\ { ( 5 a + 7 ) ( 5 a - 7 ) } \\ { ( 2 m + n ) ( 2 m - n ) } \\ { ( p + 6 q ) ( p - 6 q ) } \end{array} \right. \)

Each expression you've listed is an example of the difference of squares, which is a handy algebraic identity. This identity states that \( (a + b)(a - b) = a^2 - b^2 \). By applying this formula, we can simplify the expressions easily and recognize their structure, leading us to useful results for both factoring and expanding polynomials! For real-world applications, the difference of squares can show up in various fields, such as physics and engineering, when calculating areas or analyzing quadratic equations. For instance, if you're modeling the height of a projectile in motion, employing the difference of squares can help you derive the factors that affect its trajectory, ultimately leading to more efficient designs in things like sports equipment or building structures!