Solve for x . \( \frac{x}{x-1}-\frac{2}{x+9}=\frac{3 x+7}{x^{2}+8 x-9} \)

To solve the equation \[ \frac{x}{x-1}-\frac{2}{x+9}=\frac{3x+7}{x^{2}+8x-9}, \] first, let's simplify and combine the fractions on the left side. The common denominator for the left-hand side is \((x-1)(x+9)\). Rewriting the equation: \[ \frac{x(x+9) - 2(x-1)}{(x-1)(x+9)} = \frac{3x + 7}{x^{2} + 8x - 9}. \] Now let's simplify the numerator: \[ x(x+9) - 2(x-1) = x^2 + 9x - 2x + 2 = x^2 + 7x + 2. \] Thus, we have: \[ \frac{x^2 + 7x + 2}{(x-1)(x+9)} = \frac{3x + 7}{(x-3)(x+3)}. \] Now, we will cross-multiply: \[ (x^2 + 7x + 2)(x^2 - 9) = (3x + 7)(x^2 + 8x - 9). \] We need to expand both sides: Left Side: \[ (x^2 + 7x + 2)(x^2 - 9) = x^4 - 9x^2 + 7x^3 - 63x + 2x^2 - 18, \] which simplifies to: \[ x^4 + 7x^3 - 7x^2 - 63x - 18. \] Right Side: \[ (3x + 7)(x^2 + 8x - 9) = 3x^3 + 24x^2 - 27x + 7x^2 + 56x - 63. \] This simplifies to: \[ 3x^3 + 31x^2 + 29x - 63. \] Setting both sides equal gives us: \[ x^4 + 7x^3 - 7x^2 - 63x - 18 = 3x^3 + 31x^2 + 29x - 63. \] Rearranging this results in: \[ x^4 + 4x^3 - 38x^2 - 92x + 45 = 0. \] This is a polynomial that might need numerical methods or factoring techniques to solve, depending on its nature. You can use synthetic division or the Rational Root Theorem to find potential rational roots and ultimately solve for \( x \).