Binomials and Distribution \( 3(3 x+5 y)(2 x+3 y)-2(x+3 y)(3 x-4 y) \)

To unravel this expression, we begin by expanding each binomial. The first part, \( 3(3x + 5y)(2x + 3y) \), expands to \( 3(6x^2 + 9xy + 10xy + 15y^2) = 3(6x^2 + 19xy + 15y^2) = 18x^2 + 57xy + 45y^2 \). Next, for the second part, \( -2(x + 3y)(3x - 4y) \), we expand it as \( -2(3x^2 - 4xy + 9xy - 12y^2) = -2(3x^2 + 5xy - 12y^2) = -6x^2 - 10xy + 24y^2 \). Now, we combine both results: \[ (18x^2 + 57xy + 45y^2) + (-6x^2 - 10xy + 24y^2) \] This simplifies to: \[ 12x^2 + 47xy + 69y^2 \] So, the final simplified expression is \( 12x^2 + 47xy + 69y^2 \). Keep your distribution skills sharp—practicing with different combinations helps you become more proficient! Remember to check your signs and combine like terms to avoid any slip-ups that could lead to errors in your results.