factor \( 6 x^{4}+8 x^{2}=26 x^{3} \)

To factor the equation, first, rearrange it into standard form: \( 6x^4 - 26x^3 + 8x^2 = 0 \). Next, let's group the terms and factor out the common term. Notice that every term has a common factor of \( 2x^2 \). Factoring that out gives: \( 2x^2(3x^2 - 13x + 4) = 0 \). Now, focus on factoring the quadratic \( 3x^2 - 13x + 4 \). Look for two numbers that multiply to \( 3 \times 4 = 12 \) and add to \( -13 \). This can be achieved with \( -12 \) and \( -1 \). Rewriting gives: \( 3x^2 - 12x - x + 4 = 0 \). Now, factor by grouping: \( 3x(x - 4) - 1(x - 4) = 0 \). This results in: \( (3x - 1)(x - 4) = 0 \). Thus, the complete factorization is: \( 2x^2(3x - 1)(x - 4) = 0 \). Setting each factor equal to zero will give you the solutions for \( x \). Happy factoring!