Factor completely. $$ 100 v^{2}+220 v+121 $$

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1. Recognize that the quadratic $$100v^2 + 220v + 121$$ might be a perfect square trinomial of the form $$(av + b)^2$$.
2. Expand $$(av + b)^2$$ to get $$a^2v^2 + 2ab\,v + b^2$$.
3. Set up the system of equations by matching coefficients:
   - $$a^2 = 100$$ implies $$a = 10$$ since $$10^2 = 100$$.
   - $$b^2 = 121$$ implies $$b = 11$$ since $$11^2 = 121$$.
4. Verify the middle term: 
   - $$2ab = 2 \times 10 \times 11 = 220$$, which matches the coefficient of $$v$$.
5. Therefore, the quadratic factors as:
   $$
   100v^2 + 220v + 121 = (10v + 11)^2
   $$
$$100v^{2} + 220v + 121 = (10v + 11)^2$$

Factor completely.

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