Find the sum of all possible positive integer values of $b$ such that the quadratic equation $2x^2 + 5x + b = 0$ has rational roots.
\(\text{Rational roots will occur when the discriminant is a non-negative perfect square}\\ D=b^2 - 4ac = 25-(4)(2)(b) = 25-8b\\ \text{$D$ is a perfect square for $b=0, 2, 3$}\\ \sum b = 5\)