jmc

Because the coefficients of the polynomial are rational, the radical conjugate $-1+4\sqrt{2}$ must also be a root of the polynomial. By Vieta's formulas, the product of the roots of this polynomial is $-31,$ and the product of these two roots is $(-1-4\sqrt{2})(-1+4\sqrt{2})=-31,$ so the remaining root must be $\frac{-31}{-31}=1.$ Then by Vieta's formulas again, we have $$a=-[1+(-1-4\sqrt{2})+(-1+4\sqrt{2})]=\boxed{1}.$$