Factor \( x^{4}+x^{2}-90 \) completely. All factors in your answer should have integer coefficients.

To factor the expression \( x^{4} + x^{2} - 90 \), we can start by making a substitution. Let \( y = x^{2} \). This transforms our expression into a quadratic: \[ y^{2} + y - 90 \] Next, we need to factor the quadratic. We are looking for two numbers that multiply to \(-90\) (the constant term) and add up to \(1\) (the coefficient of \(y\)). These two numbers are \(10\) and \(-9\), since: \[ 10 \cdot (-9) = -90 \quad \text{and} \quad 10 + (-9) = 1 \] Using these numbers to factor, we have: \[ y^{2} + y - 90 = (y + 10)(y - 9) \] Now, we revert back to \(x\) by substituting \(y\) back with \(x^{2}\): \[ (x^{2} + 10)(x^{2} - 9) \] The expression \(x^{2} - 9\) can be factored further as a difference of squares: \[ x^{2} - 9 = (x - 3)(x + 3) \] Now, substituting back we have: \[ x^{4} + x^{2} - 90 = (x^{2} + 10)(x - 3)(x + 3) \] Finally, we arrive at the completely factored form of the original expression: \[ \boxed{(x^{2} + 10)(x - 3)(x + 3)} \]