Answer: A

Video solution:

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Written solution:

By Vieta's formulas, the six roots have sum $10$ and product $16.$ Because the product is a power of $2,$ every positive integer root is a power of $2.$ Distributing the four factors of $2$ among six roots gives the least possible sum when four roots are $2$ and two roots are $1$; that sum is already $10.$ Hence the roots are $$1,1,2,2,2,2.$$

The coefficient $B$ is the negative of the sum of all products of three roots. Choosing zero, one, or two of the two roots equal to $1$ gives $$\begin{aligned} B&=-\left(\binom43 2^3+2\binom42 2^2\right.\\ &\qquad\left.+\binom41 2\right)\\ &=-88. \end{aligned}$$

Thus, A is the correct answer.