Simplify \( \frac{3^{2 x}-3^{x}-6}{3^{x}-3} \)

To simplify the expression \( \frac{3^{2x}-3^{x}-6}{3^{x}-3} \), let's start by making a substitution. Let \( y = 3^{x} \). Then, we can write \( 3^{2x} = (3^{x})^2 = y^2 \). Now we can rewrite the expression: \[ \frac{y^2 - y - 6}{y - 3} \] Next, we can factor the numerator: \[ y^2 - y - 6 = (y - 3)(y + 2) \] So the expression becomes: \[ \frac{(y - 3)(y + 2)}{y - 3} \] Since \( y \neq 3 \) (we will avoid this value to prevent division by zero), we can cancel \( y - 3 \): \[ y + 2 \] Now substituting back \( y = 3^{x} \): \[ 3^{x} + 2 \] Thus, the simplified expression is: \[ 3^{x} + 2 \]