Find the value of $v$ such that $\frac{-21-\sqrt{201}}{10}$ a root of $5x^2+21x+v = 0$.
We don't actually have to evaluate the expression $\frac{-21 - \sqrt{201}}{10}$ to solve the problem. Since it is a root of the quadratic equation $5x^2 + 21x + v = 0$, we must have that $\frac{-21 - \sqrt{201}}{10}$ is equal to one of the values of the roots of the equation. But the discriminant of the quadratic equation $5x^2 + 21x + v = 0$ is $21^2 - 4 \cdot 5 \cdot v = 441 - 20v$, and the quadratic formula tells us that the roots are \begin{align*}
x=\frac{-21 \pm \sqrt{441 - 20v}}{10}.
\end{align*} Thus, we want this expression to equal $\frac{-21 - \sqrt{201}}{10}$. So we set \begin{align*}
\frac{-21 + \sqrt{441 - 20v}}{10} &= \frac{-21 - \sqrt{201}}{10} \\
-21 + \sqrt{441 - 20v} &= -21 - \sqrt{201} \\
2\sqrt{441 - 20v} &= \sqrt{201} \\
4(441 - 20v) &= (\sqrt{201})^2 \\
4(441 - 20v) &= 201 \\
1764 - 80v &= 201 \\
80v &= 1563 \\
v &= \boxed{\frac{1563}{80}}.
\end{align*}
To find the value of $v$ such that $\frac{-21-\sqrt{201}}{10}$ is a root of $5x^2+21x+v = 0$, we can use the fact that the roots of a quadratic equation can be found using the formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In this case, the quadratic equation is $5x^2+21x+v = 0$, so we have $a = 5$, $b = 21$, and $c = v$.
Since $\frac{-21-\sqrt{201}}{10}$ is a root, we can substitute this value for $x$ in the equation and solve for $v$:
\[0 = 5\left(\frac{-21-\sqrt{201}}{10}\right)^2 + 21\left(\frac{-21-\sqrt{201}}{10}\right) + v\]
Simplifying this expression:
\[0 = 5\left(\frac{441 + 42\sqrt{201} + 201}{100}\right) - 21\left(\frac{-21 - \sqrt{201}}{10}\right) + v\]
\[0 = \frac{5(642 + 42\sqrt{201}) - 21(-210 - 21\sqrt{201})}{100} + v\]
\[0 = \frac{3214 + 210\sqrt{201} + 4410 + 441\sqrt{201} + 2100 + 210\sqrt{201}}{100} + v\]
\[0 = \frac{9724 + 861\sqrt{201}}{100} + v\]
To eliminate the square root, we can equate the numerator to zero:
\[9724 + 861\sqrt{201} = 0\]
Subtracting 9724 from both sides:
\[861\sqrt{201} = -9724\]
Dividing both sides by 861:
\[\sqrt{201} = \frac{-9724}{861} = -\frac{9724}{861}\]
Now, we can square both sides to eliminate the square root:
\[\left(\sqrt{201}\right)^2 = \left(-\frac{9724}{861}\right)^2\]
\[201 = \frac{9724^2}{861^2}\]
Solving for $v$:
\[v = -\frac{9724}{100} = -97.24\]
Therefore, the value of $v$ such that $\frac{-21-\sqrt{201}}{10}$ is a root of $5x^2+21x+v = 0$ is $v = -97.24$.