Factor \( x^{2}\left(6 x^{2}+8-26 x\right)=0 \)

To factor the equation \( x^{2}(6 x^{2}+8-26 x) = 0 \), we first analyze the quadratic expression inside the parentheses. Reordering it gives us \( 6x^2 - 26x + 8 \). Next, we can factor this quadratic expression. First, we look for two numbers that multiply to \( 6 \times 8 = 48 \) and add to \( -26 \). The numbers \(-24\) and \(-2\) fit perfectly! Thus, we can rewrite the quadratic as: \[ 6x^2 - 24x - 2x + 8 = 0 \] Now, grouping gives us: \[ 6x(x - 4) - 2(x - 4) = 0 \] Factoring out \( (x - 4) \), we find: \[ (6x - 2)(x - 4) = 0 \] This means \( x^{2}(6x - 2)(x - 4) = 0 \). To finish, we can also simplify \( 6x - 2 \) by factoring out a 2: \[ x^{2} \cdot 2(3x - 1)(x - 4) = 0 \] Thus, the complete factorization is: \[ 2x^{2}(3x - 1)(x - 4) = 0 \] Now, the solutions are \( x = 0, \frac{1}{3}, 4 \).