A rational expression is undefined when the denominator is equal to zero. For the expression \(\frac{4v^2}{8v + 9}\), we need to find the values of \(v\) that make the denominator \(8v + 9\) equal to zero.
To find these values, we set the denominator equal to zero and solve for \(v\):
\[ 8v + 9 = 0 \]
Subtract \(9\) from both sides:
\[ 8v = -9 \]
Now, divide both sides by \(8\):
\[ v = -\frac{9}{8} \]
Thus, the expression \(\frac{4v^2}{8v + 9}\) is undefined when \(v = -\frac{9}{8}\).