\[x^{2} - 2xy - 63y^{2} =\]
\[= x^{2} - 2xy + y^{2} - 64y^{2} =\]
\[= (x - y)^{2} - 64y^{2} =\]
\[= (x - y - 8y)(x - y + 8y) =\]
\[= (x - 9y)(x + 7y)\]