\( \frac{3 x+1}{6} \times \frac{3}{3 x+1} \) e \( \frac{(2 x-4)(x+6)}{x+7} \times \frac{x+7}{2 x-4} \)

Simplifying the first expression: \( \frac{3 x+1}{6} \times \frac{3}{3 x+1} \) is pretty straightforward. The \(3x + 1\) in the numerator of one fraction cancels with the \(3x + 1\) in the denominator of the other, leaving you with \( \frac{3}{6} = \frac{1}{2} \). Now, onto the second part: \( \frac{(2 x-4)(x+6)}{x+7} \times \frac{x+7}{2 x-4} \). Here, the \(2x - 4\) cancels out with itself, as does the \(x + 7\). What remains is simply \(x + 6\). So, you’d be left with \(x + 6\). Both expressions showcase how elegant and powerful simplification can be!