Factor the polynomial. \[ x^{4}+4 x^{2}-117 \]

To factor the polynomial \( x^{4}+4 x^{2}-117 \), let's first make a substitution. Let \( y = x^2 \). Then, the polynomial can be rewritten as: \[ y^2 + 4y - 117 \] Now, we need to factor this quadratic polynomial. We look for two numbers that multiply to \(-117\) and add up to \(4\). Those numbers are \(13\) and \(-9\). Thus, we can factor it as: \[ (y + 13)(y - 9) \] Substituting back \(y = x^2\): \[ (x^2 + 13)(x^2 - 9) \] Next, we can factor \(x^2 - 9\) further since it is a difference of squares: \[ x^2 - 9 = (x - 3)(x + 3) \] Thus, the complete factorization of the original polynomial is: \[ (x^2 + 13)(x - 3)(x + 3) \] And that’s your fully factored polynomial!