1. **State the problem:** Factor the polynomial $$100x^2 + 180x + 81$$. 2. **Recall the formula for factoring a perfect square trinomial:** $$a^2 + 2ab + b^2 = (a + b)^2$$ 3. **Check if the polynomial fits this form:** - Identify $a^2 = 100x^2$, so $a = 10x$. - Identify $b^2 = 81$, so $b = 9$. - Check the middle term: $2ab = 2 \times 10x \times 9 = 180x$, which matches the middle term. 4. **Since the polynomial matches the perfect square trinomial form, factor it as:** $$100x^2 + 180x + 81 = (10x + 9)^2$$ 5. **Final answer:** $$\boxed{(10x + 9)^2}$$