1)) Factor \( x^{4}+12 x^{2}+27 \) completely. All factors in your answer should have integer coefficients.

To factor \( x^{4}+12 x^{2}+27 \), we can start by making a substitution to simplify the expression. Let \( y = x^{2} \). The expression then becomes \( y^{2} + 12y + 27 \). We will look for two numbers that multiply to \( 27 \) (the constant term) and add up to \( 12 \) (the coefficient of the linear term). The numbers \( 3 \) and \( 9 \) satisfy both conditions since \( 3 \cdot 9 = 27 \) and \( 3 + 9 = 12 \). Now we can factor the quadratic expression: \[ y^{2} + 12y + 27 = (y + 3)(y + 9) \] Substituting back \( y = x^{2} \), we get: \[ (x^{2} + 3)(x^{2} + 9) \] Thus, the completely factored form of the original expression is: \[ \boxed{(x^{2} + 3)(x^{2} + 9)} \]