Subtract. \[ \frac{x}{x^{2}-2 x-24}-\frac{x}{x^{2}-8 x+12} \]

To subtract the two fractions, we first need to find a common denominator. The denominators are \(x^2 - 2x - 24\) and \(x^2 - 8x + 12\). Let's factor both denominators: 1. For \(x^2 - 2x - 24\), we need two numbers that multiply to \(-24\) and add to \(-2\). These numbers are \(-6\) and \(4\). So, we can factor as: \[ x^2 - 2x - 24 = (x - 6)(x + 4) \] 2. For \(x^2 - 8x + 12\), we need two numbers that multiply to \(12\) and add to \(-8\). These numbers are \(-6\) and \(-2\). So, we can factor as: \[ x^2 - 8x + 12 = (x - 6)(x - 2) \] Now that we have the factored forms of the denominators, we can write the expression as: \[ \frac{x}{(x - 6)(x + 4)} - \frac{x}{(x - 6)(x - 2)} \] The common denominator is \((x - 6)(x + 4)(x - 2)\). We rewrite each fraction to have this common denominator: \[ \frac{x(x - 2)}{(x - 6)(x + 4)(x - 2)} - \frac{x(x + 4)}{(x - 6)(x + 4)(x - 2)} \] Now we can combine the fractions: \[ \frac{x(x - 2) - x(x + 4)}{(x - 6)(x + 4)(x - 2)} \] Now, simplify the numerator: \[ x(x - 2) - x(x + 4) = x^2 - 2x - (x^2 + 4x) = x^2 - 2x - x^2 - 4x = -6x \] Thus, we rewrite the expression: \[ \frac{-6x}{(x - 6)(x + 4)(x - 2)} \] Finally, our results yield: \[ \frac{-6x}{(x - 6)(x + 4)(x - 2)} \] This is the simplified form of the original expression.