\( \left[\frac{1}{3 x-2} \times \frac{9 x^{2}-4}{3 x^{2}-13 x-10}\right] \frac{7}{x-1} \) as a single fraction in its simplest form.

To simplify the expression \( \left[\frac{1}{3 x-2} \times \frac{9 x^{2}-4}{3 x^{2}-13 x-10}\right] \frac{7}{x-1} \), we will follow a few steps. First, factor \( 9x^2 - 4 \) and \( 3x^2 - 13x - 10 \): 1. \( 9x^2 - 4 \) is a difference of squares: \[ 9x^2 - 4 = (3x - 2)(3x + 2) \] 2. To factor \( 3x^2 - 13x - 10 \), we look for two numbers that multiply to \( 3 \cdot (-10) = -30 \) and sum to \( -13 \). These numbers are \( -15 \) and \( 2 \): \[ 3x^2 - 15x + 2x - 10 = 3x(x - 5) + 2(x - 5) = (3x + 2)(x - 5) \] Now substituting back into the original expression: \[ \left[\frac{1}{3x-2} \times \frac{(3x - 2)(3x + 2)}{(3x + 2)(x - 5)}\right] \frac{7}{x - 1} \] Cancelling \( 3x - 2 \) gives: \[ \frac{(3x + 2)}{(x - 5)} \cdot \frac{7}{(x - 1)} \] This leads to the expression: \[ \frac{7(3x + 2)}{(x - 5)(x - 1)} \] Thus, the final simplified expression is: \[ \frac{7(3x + 2)}{(x - 5)(x - 1)} \]