1. **State the problem:** We want to factor the expression $$3^{2x} + 3^x - 6$$ into the form $$(3^x + a)(3^x + b)$$ and find the values of $a$ and $b$. 2. **Rewrite the expression:** Let $$y = 3^x$$. Then the expression becomes $$y^2 + y - 6$$. 3. **Factor the quadratic:** We want to find $a$ and $b$ such that: $$y^2 + y - 6 = (y + a)(y + b) = y^2 + (a + b)y + ab$$ 4. **Match coefficients:** From the equation above, we get two conditions: - Sum of $a$ and $b$: $$a + b = 1$$ - Product of $a$ and $b$: $$ab = -6$$ 5. **Find $a$ and $b$:** We look for two numbers that multiply to $-6$ and add to $1$. 6. **Check possible pairs:** - $3$ and $-2$: $3 + (-2) = 1$ and $3 \times (-2) = -6$ 7. **Conclusion:** The values are $$a = 3$$ and $$b = -2$$. 8. **Write the factorization:** $$3^{2x} + 3^x - 6 = (3^x + 3)(3^x - 2)$$. **Final answer:** $a = 3$, $b = -2$ which corresponds to option 4 (a=2, b=-3) is incorrect, so the correct pair is $a=3$, $b=-2$ which is not listed exactly but closest to option 2 (a=-2, b=3) if order is swapped. Since multiplication is commutative, option 2 is correct.