Solve for \( u \). \[ \frac{6}{u-4}+\frac{2}{u^{2}-16}=\frac{1}{u+4} \]

To solve the equation \[ \frac{6}{u-4} + \frac{2}{u^{2}-16} = \frac{1}{u+4}, \] we start by factoring the denominator in the second term: \[ u^2 - 16 = (u-4)(u+4). \] Thus, we can rewrite the equation as: \[ \frac{6}{u-4} + \frac{2}{(u-4)(u+4)} = \frac{1}{u+4}. \] Next, to eliminate the fractions, we can multiply through by \((u-4)(u+4)\): \[ 6(u+4) + 2 = (u-4). \] Now distribute: \[ 6u + 24 + 2 = u - 4. \] This simplifies to: \[ 6u + 26 = u - 4. \] Next, we can move all terms involving \(u\) to one side and constants to the other side: \[ 6u - u = -4 - 26. \] This simplifies to: \[ 5u = -30. \] Now dividing both sides by 5 gives us: \[ u = -6. \] Finally, we should check if our solution satisfies the original equation. Substitute \(u = -6\) back into the original equation: 1. Calculate \(u - 4 = -6 - 4 = -10\), 2. Calculate \(u^2 - 16 = (-6)^2 - 16 = 36 - 16 = 20\), 3. Calculate \(u + 4 = -6 + 4 = -2\). Now substituting into the fractions: \[ \frac{6}{-10} + \frac{2}{20} = \frac{1}{-2}. \] This evaluates to: \[ -\frac{3}{5} + \frac{1}{10} = -\frac{1}{2}. \] Finding a common denominator for the left-hand side: \[ -\frac{3}{5} = -\frac{6}{10}, \] thus: \[ -\frac{6}{10} + \frac{1}{10} = -\frac{5}{10} = -\frac{1}{2}. \] Since both sides are equal, our solution is verified. Therefore, the solution is \[ \boxed{-6}. \]