sat-dog

The correct answer is $8$. Since each term of the given expression, $2x^3+42x^2+208x$, has a factor of $2x$, the expression can be rewritten as $2x\left(x^2\right)+2x\left(21x\right)+2x\left(104\right)$, or $2x\left(x^2+21x+104\right)$. Since the values $8$ and $13$ have a sum of $21$ and a product of $104$, the expression $x^2+21x+104$ can be factored as $\left(x+8\right)\left(x+13\right)$. Therefore, the given expression can be factored as $2x\left(x+8\right)\left(x+13\right)$. It follows that the factors of the given expression are $2$, $x$, $x+8$, and $x+13$. Of these factors, only $x+8$ and $x+13$ are of the form $x+b$, where $b$ is a positive constant. Therefore, the possible values of $b$ are $8$ and $13$. Thus, the smallest possible value of $b$ is $8$.