)) Factor \( x^{4}+14 x^{2}+49 \) completely. All factors in your answer should have integer coefficients. Sideo

To factor \( x^{4}+14 x^{2}+49 \) completely, we can start by recognizing that the expression can be treated as a quadratic in terms of \( x^2 \). Let's set \( y = x^2 \), transforming the expression into \( y^2 + 14y + 49 \). This can be factored as \( (y + 7)(y + 7) \) or \( (y + 7)^2 \). Substituting back \( y = x^2 \) gives us \( (x^2 + 7)^2 \). Thus, the complete factorization of the original polynomial is: \[ (x^2 + 7)^2 \] And there you have it—factor the quadratic first to reveal its perfect square nature! For a deeper understanding, remember that recognizing patterns can simplify your factoring journey. Always look for perfect squares or sum of squares. Moreover, ensure to check the degree of polynomials when applying these techniques; minor miscalculations can lead to frustrating dead ends! So, keep your factoring skills sharp—being familiar with common structures can significantly speed up your work.