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Choice B is correct. The given expression has a common factor of $2$ in the denominator, so the expression can be rewritten as $\frac{8x\left(x-7\right)-3\left(x-7\right)}{2\left(x-7\right)}$. The three terms in this expression have a common factor of $\left(x-7\right)$. Since it's given that $x>7$, $x$ can't be equal to $7$, which means $\left(x-7\right)$ can't be equal to $0$. Therefore, each term in the expression, $\frac{8x\left(x-7\right)-3\left(x-7\right)}{2\left(x-7\right)}$, can be divided by $\left(x-7\right)$, which gives $\frac{8x-3}{2}$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.