Solve the following equation by factoring. \( \frac{7(x-2)}{x-3}+\frac{2}{x}=\frac{-2}{x(x-3)} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \} \) (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There is no solution.

To solve the equation \( \frac{7(x-2)}{x-3}+\frac{2}{x}=\frac{-2}{x(x-3)} \), we first eliminate the fractions by multiplying every term by the common denominator \( x(x-3) \). This gives us: \[ 7(x-2)x + 2(x-3) = -2. \] Expanding and simplifying leads to a quadratic equation. Factoring the quadratic will yield the potential solutions. Ensure to check each solution against the original equation, considering restrictions where \( x \neq 0 \) and \( x \neq 3 \). Once we find the solutions, we'll realize, however, that it may lead us to a situation where the defined values do not permit certain solutions. Substitute back into the original equation to verify. Here is the fun twist: Keep in mind, sometimes, when it seems like solutions are being thrown your way like confetti, they're just distractions! Once everything is factored and resolved, if a valid solution exists without contradictions, that's your winner! Otherwise, it might lead to unexpected outcomes, like finding there are no solutions at all!