1. **State the problem:** Divide the polynomial expression $$6 v^3 - 47 v^2 + 11 v + 20$$ by the binomial $$6 v - 5$$. 2. **Recall the division formula:** Polynomial division is similar to long division with numbers. We divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the divisor by that quotient term, subtract, and repeat. 3. **Divide the leading terms:** $$\frac{6 v^3}{6 v} = v^2$$. 4. **Multiply and subtract:** Multiply $$v^2$$ by $$6 v - 5$$ to get $$6 v^3 - 5 v^2$$. Subtract this from the original polynomial: $$\left(6 v^3 - 47 v^2 + 11 v + 20\right) - \left(6 v^3 - 5 v^2\right) = -42 v^2 + 11 v + 20$$. 5. **Divide the new leading term:** $$\frac{-42 v^2}{6 v} = -7 v$$. 6. **Multiply and subtract:** Multiply $$-7 v$$ by $$6 v - 5$$ to get $$-42 v^2 + 35 v$$. Subtract: $$\left(-42 v^2 + 11 v + 20\right) - \left(-42 v^2 + 35 v\right) = -24 v + 20$$. 7. **Divide the new leading term:** $$\frac{-24 v}{6 v} = -4$$. 8. **Multiply and subtract:** Multiply $$-4$$ by $$6 v - 5$$ to get $$-24 v + 20$$. Subtract: $$\left(-24 v + 20\right) - \left(-24 v + 20\right) = 0$$. 9. **Result:** The quotient is $$v^2 - 7 v - 4$$ with no remainder. **Final answer:** $$v^2 - 7 v - 4$$. This matches the first option.