### Instruction: For what real value of $v$ is $\frac{-21-\sqrt{301}}{10}$ a root of $5x^2+21x+v$?### Response: I notice that the equation is quadratic, so it must have two roots.
I also notice that the given root has a negative sign and a square root in it, which makes me think of using the quadratic formula.
If I compare the given root with the formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=5$, $b=21$, and $c=v$, I can try to match the terms and find $v$.
I see that the numerator of the given root has $-21$ and $-\sqrt{301}$, which means that $b=21$ and $b^2-4ac=301$.
I can plug in $a=5$ and $b=21$ into the second equation and get $21^2-4(5)v=301$.
Simplifying, I get $441-20v=301$, or $20v=140$, or $v=7$.
This is the value of $v$ that makes $\frac{-21-\sqrt{301}}{10}$ a root of the equation.