jmc

We see that $2x^3+bx+7$ must be the product of $x^2+px+1$ and a linear factor. Furthermore, this linear factor must be $2x+7,$ to make the cubic and constant coefficients match. Thus, $$(2x^3+bx+7)=(x^2+px+1)(2x+7).$$Expanding, we get $$2x^3+bx+7=2x^3+(2p+7)x^2+(7p+2)x+7.$$Then $2p+7=0$ and $7p+2=b.$ Solving, we find $p=-\frac{7}{2}$ and $b=\boxed{-\frac{45}{2}}.$