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Choice C is correct. The left-hand side of the given equation can be factored as $\left(5x+3\right)\left(x-8\right)$. Therefore, the given equation, $5x^2-37x-24=0$, can be written as $\left(5x+3\right)\left(x-8\right)=0$. Applying the zero product property to this equation yields $5x+3=0$ and $x-8=0$. Subtracting $3$ from both sides of the equation $5x+3=0$ yields $5x=-3$. Dividing both sides of this equation by $5$ yields $x=-\frac{3}{5}$. Adding $8$ to both sides of the equation $x-8=0$ yields $x=8$. Therefore, the two solutions to the given equation, $5x^2-37x-24=0$, are $-\frac{3}{5}$ and $8$. It follows that $8$ is the positive solution to the given equation.

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

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

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