Find the least common denominator of \( \frac{7 x}{2 x^{2}-13 x+6} \) and \( \frac{3}{2 x^{2}-5 x+2} \). \( \square \)

To find the least common denominator (LCD) of the two fractions, we first need to factor the denominators. Starting with the first denominator \( 2x^2 - 13x + 6 \), we can factor it: 1. Multiply the coefficient of \( x^2 \) (which is 2) by the constant term (which is 6), giving us \( 2 \times 6 = 12 \). 2. We need two numbers that multiply to 12 and add up to -13. These numbers are -12 and -1. 3. Rewrite the middle term: \[ 2x^2 - 12x - x + 6 = 0 \] 4. Now factor by grouping: \[ 2x(x - 6) - 1(x - 6) = (2x - 1)(x - 6) \] Next, we factor the second denominator \( 2x^2 - 5x + 2 \): 1. Multiply the coefficient of \( x^2 \) (which is 2) by the constant term (which is 2), resulting in \( 2 \times 2 = 4 \). 2. We need two numbers that multiply to 4 and add up to -5, which are -4 and -1. 3. Rewrite the middle term: \[ 2x^2 - 4x - x + 2 = 0 \] 4. Factor by grouping: \[ 2x(x - 2) - 1(x - 2) = (2x - 1)(x - 2) \] Now we have factored both denominators: - First denominator: \( (2x - 1)(x - 6) \) - Second denominator: \( (2x - 1)(x - 2) \) The least common denominator is found by taking each factor at its highest power from both factorizations: \[ \text{LCD} = (2x - 1)(x - 6)(x - 2) \] Thus, the least common denominator of the two fractions is: \[ \boxed{(2x - 1)(x - 6)(x - 2)} \]