Solve: $\frac{x^{2}-x-6}{x^{2}}=\frac{x-6}{2 x}+\frac{2 x+12}{x}$ After multiplying each side of the equation by the LCD and simplifying, the resulting equation is

Step 1 ：First, we need to eliminate the fractions in the equation. To do this, we multiply each term by the least common denominator (LCD), which is \(2x^2\).

Step 2 ：After multiplying each term by the LCD and simplifying, we get the equation \(2x^2 - 2x - 12 = x(x - 6) + 2x(2x + 12)\).

Step 3 ：Solving this equation gives us the solutions \(x = -6\) and \(x = -\frac{2}{3}\).

Step 4 ：However, we need to check these solutions in the original equation to make sure they are valid. If substituting the solution back into the original equation results in a true statement, then the solution is valid. If not, then the solution is extraneous and should be discarded.

Step 5 ：Substituting the solutions back into the original equation, we find that both solutions, \(x = -6\) and \(x = -\frac{2}{3}\), are valid as they satisfy the original equation.

Step 6 ：Final Answer: The solutions to the equation are \(\boxed{x = -6}\) and \(\boxed{x = -\frac{2}{3}}\).