Solve the equation \( \frac{x+1}{x-1}=\frac{-11}{x+3}+\frac{8}{x^{2}+2 x-3} \) List all valid solutions, separated by commas. \( x=\square \)

To solve the equation \[ \frac{x+1}{x-1}=\frac{-11}{x+3}+\frac{8}{x^{2}+2x-3}, \] we first factor the denominator of the third term on the right side. The expression \( x^2 + 2x - 3 \) can be factored as \( (x+3)(x-1) \). Thus, we can rewrite the equation as: \[ \frac{x+1}{x-1} = \frac{-11}{x+3} + \frac{8}{(x+3)(x-1)}. \] Next, we need to combine the terms on the right-hand side: \[ \frac{-11(x-1) + 8}{(x+3)(x-1)} = \frac{-11x + 11 + 8}{(x+3)(x-1)} = \frac{-11x + 19}{(x+3)(x-1)}. \] Now we have: \[ \frac{x+1}{x-1} = \frac{-11x + 19}{(x+3)(x-1)}. \] Cross-multiply to eliminate the fractions: \[ (x + 1)(x + 3) = (-11x + 19)(x - 1). \] Expanding both sides gives: \[ x^2 + 4x + 3 = -11x^2 + 11x + 19. \] Now, combine like terms: \[ 12x^2 - 7x - 16 = 0. \] Next, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 12, b = -7, c = -16 \). Calculate the discriminant: \[ b^2 - 4ac = (-7)^2 - 4(12)(-16) = 49 + 768 = 817. \] Then substituting into the formula gives: \[ x = \frac{7 \pm \sqrt{817}}{24}. \] We have two potential solutions: \[ x = \frac{7 + \sqrt{817}}{24}, \quad x = \frac{7 - \sqrt{817}}{24}. \] We should also verify that these solutions do not violate any restrictions from the original equation. The values that would make the denominator zero are: - \(x - 1 = 0 \Rightarrow x = 1\) - \(x + 3 = 0 \Rightarrow x = -3\) Neither of the founding solutions equal 1 or -3, so they are valid. Thus, the solutions are: \[ x = \frac{7 + \sqrt{817}}{24}, \frac{7 - \sqrt{817}}{24}. \] Final answer: \( x = \frac{7 + \sqrt{817}}{24}, \frac{7 - \sqrt{817}}{24} \)