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

To tackle these expressions, you can apply the distributive property (also known as the FOIL method for binomials). Let’s break it down, starting with the first expression \( (3a + 2b)(4a - 3b) \). When you multiply these, you would do: - First: \( 3a \times 4a = 12a^2 \) - Outer: \( 3a \times -3b = -9ab \) - Inner: \( 2b \times 4a = 8ab \) - Last: \( 2b \times -3b = -6b^2 \) Combining those gives: \( 12a^2 - ab - 6b^2 \). For the second expression \( (2x - 3y)(5x + 2y) \), follow the same steps: - First: \( 2x \times 5x = 10x^2 \) - Outer: \( 2x \times 2y = 4xy \) - Inner: \( -3y \times 5x = -15xy \) - Last: \( -3y \times 2y = -6y^2 \) Combining results yields: \( 10x^2 - 11xy - 6y^2 \). Now, the expression \( (4x - y)(2x + 3y) \): - First: \( 4x \times 2x = 8x^2 \) - Outer: \( 4x \times 3y = 12xy \) - Inner: \( -y \times 2x = -2xy \) - Last: \( -y \times 3y = -3y^2 \) Combining gives: \( 8x^2 + 10xy - 3y^2 \). Lastly, for \( (5x - 2y)(3x - 2y) \): - First: \( 5x \times 3x = 15x^2 \) - Outer: \( 5x \times -2y = -10xy \) - Inner: \( -2y \times 3x = -6xy \) - Last: \( -2y \times -2y = 4y^2 \) Combining results yields: \( 15x^2 - 16xy + 4y^2 \). Each of these expressions gives you insight into polynomial multiplication and how to collect similar terms to simplify the output. Keep practicing, and soon these binomials will be cakewalks!