Discussion
One of the factors of \(81x^{2} - 126xy + 49y^{2} - (5x + 3y)^{2}\) is
Answer: C
One factor is 7x - 2y.
The first three terms form a perfect square, so the expression is a difference of two squares:
\[ \begin{aligned} &(9x - 7y)^{2} - (5x + 3y)^{2} \\ &= (9x - 7y - 5x - 3y)(9x - 7y + 5x + 3y) \\ &= (4x - 10y)(14x - 4y) \\ &= 4(2x - 5y)(7x - 2y) \end{aligned} \]
So the factors are 2x - 5y and 7x - 2y. Options A and B have the right numbers with the wrong sign (7x + 2y, 2x + 5y), and 5x - 7y does not appear at all.
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