Evaluate the indefinite integral by using the substitution $u=x^{2}+6$ to reduce the integral to standard form. \[ \int 2 x\left(x^{2}+6\right)^{-3} d x \]

Step 1 ：Given the integral \(\int 2x(x^{2}+6)^{-3} dx\), we can use the method of substitution to simplify it.

Step 2 ：Let's substitute \(u = x^2 + 6\). The derivative of \(u\) with respect to \(x\) is \(du = 2x dx\).

Step 3 ：This simplifies the integral to \(\int du/u^3\), which is a standard form that can be easily integrated.

Step 4 ：The antiderivative of \(1/u^3\) is \(-2/u^2\).

Step 5 ：Substituting back for \(u\), we get \(-2/(x^2 + 6)^2\).

Step 6 ：Adding the constant of integration, we get the final answer: \(\boxed{-\frac{2}{(x^2 + 6)^2} + C}\)