2) \( \frac{9^{x}-3^{x}-6}{3^{x}-3} \)

We start with the expression   (9^x – 3^x – 6) / (3^x – 3). Step 1. Notice that 9^x = (3^2)^x = 3^(2x). To simplify the expression, let   y = 3^x. Then we have:   9^x = y^2  and  3^x = y. Substituting these into the expression gives:   (y^2 – y – 6) / (y – 3). Step 2. Factor the numerator y^2 – y – 6. Look for two numbers that multiply to –6 and add to –1. The numbers –3 and 2 work, so:   y^2 – y – 6 = (y – 3)(y + 2). Step 3. Now the expression becomes:   [(y – 3)(y + 2)] / (y – 3). Assuming y ≠ 3 (which is equivalent to 3^x ≠ 3, or x ≠ 1), we can cancel the (y – 3) factor:   = y + 2. Step 4. Substitute back y = 3^x:   = 3^x + 2. Thus, for x such that 3^x ≠ 3 (or x ≠ 1), the simplified form of the expression is:   3^x + 2. Note: When x = 1, the denominator 3^1 – 3 becomes 0, so the original expression is undefined at x = 1.