jmc

If $2x^2+5x+b=0$ has two rational solutions, then its discriminant, $5^2-4\cdot 2\cdot b=25-8b$, must be a perfect square. Since $b$ is positive, it follows that $25-8b\ge 0⟹b\in \{1,2,3\}$. Checking each one, we see that $b=2$ and $b=3$ indeed work, and their sum is $2+3=\boxed{5}$.