Factor. \[ x^{2}-9 x y-36 y^{2} \]

To factor the expression \( x^{2} - 9xy - 36y^{2} \), we look for two numbers that multiply to give \(-36y^2\) (the product of the last term) and add up to \(-9y\) (the coefficient of the middle term). The numbers \(-12y\) and \(3y\) fit this requirement because: \(-12y \cdot 3y = -36y^2\) and \(-12y + 3y = -9y\). Now, we rewrite the expression by using these two numbers: \[ x^{2} - 12xy + 3xy - 36y^{2}. \] Next, we can group the terms: \[ (x^{2} - 12xy) + (3xy - 36y^{2}). \] Factoring out common terms from each group gives: \[ x(x - 12y) + 3y(x - 12y). \] Now, we can factor out the common factor \((x - 12y)\): \[ (x - 12y)(x + 3y). \] So, the fully factored form of the expression is: \[ (x - 12y)(x + 3y). \]