Factor \( x^{4}+19 x^{2}+90 \) completely. All factors in your answer should have integer coefficients.

To factor \( x^{4} + 19x^{2} + 90 \), we can start by making a substitution to simplify the expression. Let \( y = x^{2} \). The expression then becomes: \[ y^{2} + 19y + 90 \] Next, we need to factor the quadratic \( y^{2} + 19y + 90 \). To do this, we look for two numbers that multiply to \( 90 \) (the constant term) and add to \( 19 \) (the coefficient of the linear term). The numbers \( 9 \) and \( 10 \) fit these criteria since: \[ 9 \times 10 = 90 \quad \text{and} \quad 9 + 10 = 19 \] Thus, we can write: \[ y^{2} + 19y + 90 = (y + 9)(y + 10) \] Now, substituting back \( y = x^{2} \): \[ (x^{2} + 9)(x^{2} + 10) \] Both \( x^{2} + 9 \) and \( x^{2} + 10 \) do not factor further over the integers, as they do not have real roots. Thus, the completely factored form of \( x^{4} + 19x^{2} + 90 \) is: \[ \boxed{(x^{2} + 9)(x^{2} + 10)} \]