Solve the equation. \[ \frac{3 x+1}{x-2}+\frac{4}{x}=\frac{-8}{x^{2}-2 x} \]

Step 1 ：First, we need to get rid of the fractions in the equation \(\frac{3 x+1}{x-2}+\frac{4}{x}=-\frac{8}{x^{2}-2 x}\). We can do this by multiplying each term by \(x(x-2)\), which is the least common multiple of the denominators. This will give us a quadratic equation that we can solve.

Step 2 ：Solving the quadratic equation, we find the solutions to be \(x = -\frac{5}{3}\) and \(x = 0\).

Step 3 ：However, we need to check these solutions in the original equation because they may not be valid if they make the denominator zero.

Step 4 ：After checking, we find that the only valid solution is \(x = -\frac{5}{3}\).

Step 5 ：Final Answer: The only valid solution to the equation is \(\boxed{x = -\frac{5}{3}}\).