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Choice B is correct. The area of a rectangle is equal to its length multiplied by its width. Multiplying the given length, $x$ units, by the given width, $\left(x-15\right)$ units, yields $x\left(x-15\right)$ square units. If the rectangle has an area of $76$ square units, it follows that $x\left(x-15\right)=76$, or $x^2-15x=76$. Subtracting $76$ from both sides of this equation yields $x^2-15x-76=0$. Factoring the left-hand side of this equation yields $\left(x-19\right)\left(x+4\right)=0$. Applying the zero product property to this equation yields two solutions: $x=19$ and $x=-4$. Since $x$ is the rectangle’s length, in units, which must be positive, the value of $x$ is $19$.

Choice A is incorrect. This is the width, in units, of the rectangle, not the value of $x$.

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

Choice D is incorrect. This is the area, in square units, of the rectangle, not the value of $x$.