\( \left. \begin{array} { l } { ( 4 x + 3 y ) ( 2 x + y ) = } \\ { ( 3 x + 2 y ) ( 5 x + 2 y ) = } \\ { ( 5 a + 3 b ) ( 2 a + 3 b ) = } \\ { ( 2 x - 5 y ) ( 3 x - 4 y ) = } \end{array} \right. \)

Let’s delve into some interesting details! First off, the expression \( (4x + 3y)(2x + y) \) is like a little math factory producing a polynomial! When we expand it using the distributive property, we combine like terms to reveal a delightful quadratic in two variables, offering a glimpse into how algebra intertwines relationships between x and y. Now let’s take the expression \( (3x + 2y)(5x + 2y) \). This is another classic example of applying the distributive property, where each term multiplies by each other, leading us to a polynomial that can be visualized as a geometric area if we interpret x and y as dimensions. This not only applies in theory but is a practical method for modeling real-world scenarios like calculating areas in fields such as engineering or economics!