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

To factor \( x^{4} + 4x^{2} + 4 \) completely, let's start by recognizing that this expression can be rewritten by substituting \( y = x^{2} \), which gives us \( y^{2} + 4y + 4 \). This is a quadratic expression that can be factored as \( (y + 2)^{2} \). Now, substituting \( y \) back in gives us: \[ (x^{2} + 2)^{2} \] Thus, the completely factored form of the expression \( x^{4} + 4x^{2} + 4 \) with integer coefficients is: \[ (x^{2} + 2)(x^{2} + 2) \] or simply \((x^{2} + 2)^{2}\).