jmc

For this problem, we make use of the correspondence between sums/products of roots and coefficients of a polynomial.

Denote the two roots of the equation $\alpha$ and $\beta$. We know that $\alpha \beta =18,$ and $\alpha /\beta =2⟹\alpha =2\beta .$

So $b=-\alpha -\beta =-3\beta .$ To maximize $b,$ we want to make $\beta$ negative and as large as possible. Given the relationship that $\alpha =2\beta ,$ we see that $\beta =3$ or $-3.$ Clearly $-3$ maximizes $b,$ and $b=\boxed{9}.$