Let $b$ be a number such that $(2b+5)(b-1)=6b.$ What is the largest possible value of $b$? Express your answer as a common fraction.
We proceed as follows: \begin{align*}
(2b + 5)(b - 1) &= 6b\\
2b^2 + 3b - 5 &= 6b\\
2b^2 - 3b - 5 &= 0\\
(2b - 5)(b + 1) &= 0.
\end{align*}This gives us $b = \frac{5}{2}$ or $b = -1.$ Of these, $\boxed{\frac{5}{2}}$ is the larger possible value of $b.$