\( \left. \begin{array} { l } { ( x - 5 ) ( x - 2 ) } \\ { ( 2 x - 1 ) ( 2 x + 3 ) } \end{array} \right. \)

When you multiply polynomials like \( (x - 5)(x - 2) \), you're implementing the distributive property, also known as the FOIL method (First, Outside, Inside, Last). So, \( (x - 5)(x - 2) \) expands to \( x^2 - 7x + 10 \). For \( (2x - 1)(2x + 3) \), you would get \( 4x^2 + 6x - 2x - 3 \), which simplifies to \( 4x^2 + 4x - 3 \). Voila! You've mastered polynomial multiplication. Polynomials are used in various real-world applications like physics, engineering, and economics to model different behaviors and trends. For instance, a quadratic polynomial can represent the trajectory of an object in projectile motion, showing how height changes over distance. Similarly, in business, polynomial regression can help predict sales trends, allowing companies to make informed decisions. So, those equations aren’t just numbers; they can be powerful tools!