Factor \( x^{2}\left(6 x^{2}-26 x^{3}+8\right) \)

To factor the expression \( x^{2}\left(6 x^{2}-26 x^{3}+8\right) \), we start by focusing on the quadratic expression inside the parentheses: \( 6x^{2} - 26x^{3} + 8 \). Rearranging it, we can write it as \( -26x^{3} + 6x^{2} + 8 \). Now, we look for common factors and try factoring it by grouping or using the quadratic formula. However, first, we can rewrite \( -26x^{3} + 6x^{2} + 8 \) as \( -2(13x^{3} - 3x^{2} - 4) \). To help us factor \( 13x^{3} - 3x^{2} - 4 \), we can use the Rational Root Theorem. Testing values, we find that \( x = 1 \) is a root. After performing synthetic division, we can factor \( 13x^{3} - 3x^{2} - 4 \) as \( (x - 1)(13x^{2} + 10x + 4) \), which can be further factored if we can find rational roots for the quadratic. Thus, the complete factorization gives us: \[ x^{2} \cdot -2(x-1)(13x^{2} + 10x + 4) \] In summary, the factored form is: \[ -2x^{2}(x - 1)(13x^{2} + 10x + 4) \]