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 $\alpha \beta =48$, and $\alpha /\beta =3⟹\alpha =3\beta$.

So $b=-\alpha -\beta =-4\beta$. To maximize $b$, we want to make $\beta$ negative and as large as possible. Given the relationship that $\alpha =3\beta$ and that $\alpha \ast \beta =48$, we see that $\beta =4$ or $-4$. Clearly $-4$ maximizes $b$, and $b=\boxed{16}$.