If $a+b=k$ then $a = k-b$. The equation with solution $-b$ is $x^3 + 9x^2 + 28x + 31 = 0$. Comparing this with the equation for $a$, it looks as though it would be best to write this in terms of $x-1$. Then it becomes $(x-1)^3 + 12(x-1)^2 + 49(x-1) + 69 = 0$. That is exactly the equation satisfied by $a$. So with $x = -b$ and $x-1 = a$ it follows that $-b-1=a$, hence $a+b = -1$.