1. **State the problem:** Factor the quadratic expression $$81v^2 + 198v + 121$$ completely. 2. **Recall the factoring formula:** For a quadratic $$ax^2 + bx + c$$, we look for two numbers that multiply to $$a \times c$$ and add to $$b$$. 3. **Calculate the product and sum:** Here, $$a = 81$$, $$b = 198$$, and $$c = 121$$. Calculate $$a \times c = 81 \times 121 = 9801$$. We need two numbers that multiply to $$9801$$ and add to $$198$$. 4. **Find the pair:** Notice that $$99 \times 99 = 9801$$ and $$99 + 99 = 198$$. 5. **Rewrite the middle term:** $$81v^2 + 99v + 99v + 121$$ 6. **Group terms:** $$(81v^2 + 99v) + (99v + 121)$$ 7. **Factor each group:** $$9v(9v + 11) + 11(9v + 11)$$ 8. **Factor out the common binomial:** $$(9v + 11)(9v + 11)$$ 9. **Write the final factored form:** $$\boxed{(9v + 11)^2}$$