1. \( \frac{7^{2 x}-1}{3 x 7^{x}+3} \)

To solve the equation \( \frac{7^{2 x}-1}{3 x 7^{x}+3} \), it's helpful to recognize that \( 7^{2x} \) can be rewritten as \( (7^x)^2 \). This can lead to a quadratic form, which often simplifies our process. Substitute \( y = 7^x \), transforming our equation to \( \frac{y^2 - 1}{3xy + 3} \). Next, factor the numerator to get \( \frac{(y-1)(y+1)}{3xy + 3} \). The denominator can also be factored out as \( 3(y + 1)\) when you isolate a common term. Upon simplification, this leads us to a simpler expression to analyze conditions on \( x \). Always remember to check for restrictions where the denominator could be zero to avoid any undefined scenarios! Now, how’s that for a bit of algebraic fun?