Factor \( x^{4}-13 x^{2}+36 \) completely. All factors in your answer should have integer coefficients.

To factor the expression \( x^{4}-13x^{2}+36 \), we can start by letting \( y = x^{2} \). This transforms our expression into a quadratic form: \[ y^{2} - 13y + 36 \] Next, we want to factor this quadratic. We need to find two numbers that multiply to \( 36 \) (the constant term) and add up to \( -13 \) (the coefficient of \( y \)). The numbers that fit this requirement are \( -4 \) and \( -9 \): \[ y^{2} - 4y - 9y + 36 = (y - 4)(y - 9) \] Now we can express this back in terms of \( x \): \[ (x^{2} - 4)(x^{2} - 9) \] Both factors can be further factored: 1. \( x^{2} - 4 \) is a difference of squares: \[ x^{2} - 4 = (x - 2)(x + 2) \] 2. \( x^{2} - 9 \) is also a difference of squares: \[ x^{2} - 9 = (x - 3)(x + 3) \] Putting it all together, we have: \[ x^{4} - 13x^{2} + 36 = (x - 2)(x + 2)(x - 3)(x + 3) \] Thus, the complete factorization is: \[ \boxed{(x - 2)(x + 2)(x - 3)(x + 3)} \]