\(\frac{4}{x - 1} + \frac{6}{x + 1} = \frac{-12}{x^{2} - 1}\)
\(\frac{4(x + 1)}{(x - 1)(x + 1)} + \frac{6(x - 1)}{(x - 1)(x + 1)} = \frac{-12}{x^{2} - 1}\)
\(\frac{4x + 4}{x^{2} - 1} + \frac{6x - 6}{x^{2} - 1} = \frac{-12}{x^{2} - 1}\)
\(\frac{10x - 2}{x^{2} - 1} = \frac{-12}{x^{2} - 1}\)
\((10x - 2)(x^{2} - 1) = -12(x^{2} - 1)\)
\(10x^{3} - 10x^{2} - 2x^{2} + 2 = -12(x^{2}) + 12(1)\)
\(10x^{3} - 12x^{2} + 2 = -12x^{2} + 12\)
\(10x^{3} - 12x^{2} + 12x^{2} + 2 = -12x^{2} + 12x^{2} + 12\)
\(10x^{3} + 2 - 2 = 12 - 2\)
\(\frac{10x^{3}}{10} = \frac{10}{10}\)
\(x^{3} = 1\)
\(x = 1\)
