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Question
Simplify:
(3x + 2y)(4x + 3y) − (2x − y)(7x − 3y)
Solution
To simplify, we will proceed as follows:
\[\left( 3x + 2y \right)\left( 4x + 3y \right) - \left( 2x - y \right)\left( 7x - 3y \right)\]
\[ = \left[ \left( 3x + 2y \right)\left( 4x + 3y \right) \right] - \left[ \left( 2x - y \right)\left( 7x - 3y \right) \right]\]
\[= \left[ 3x\left( 4x + 3y \right) + 2y\left( 4x + 3y \right) \right] - \left[ 2x\left( 7x - 3y \right) - y\left( 7x - 3y \right) \right]\] (Distributive law)
\[= 12 x^2 + 9xy + 8xy + 6 y^2 - \left[ 14 x^2 - 6xy - 7xy + 3 y^2 \right]\]
\[ = 12 x^2 + 9xy + 8xy + 6 y^2 - 14 x^2 + 6xy + 7xy - 3 y^2\]
\[= 12 x^2 - 14 x^2 + 9xy + 8xy + 6xy + 7xy + 6 y^2 - 3 y^2\] (Rearranging)
\[= - 2 x^2 + 30xy + 3 y^2\] (Combining like terms)
Thus, the answer is \[- 2 x^2 + 30xy + 3 y^2\].
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