Discussion
One of the factors of \(4x^{2} + 9y^{2} + 12xy - 5(2x + 3y) - 14\) is :
Answer: A
One factor is \(2x + 3y - 7\).
The first three terms form a perfect square: \(4x^2 + 12xy + 9y^2 = (2x + 3y)^2\). Put \(t = 2x + 3y\).
\[ \begin{aligned} t^2 - 5t - 14 &= (t - 7)(t + 2) \\ &= (2x + 3y - 7)(2x + 3y + 2) \end{aligned} \]
So the factors are \(2x + 3y - 7\) and \(2x + 3y + 2\). Check: \(-7 \times 2 = -14\) and \(-7 + 2 = -5\). The options with + 7, + 14 and − 2 have the wrong signs or numbers.
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