Solve $LaTeX: \displaystyle \frac{x}{x + 7} - \frac{1}{x + 5}=- \frac{2}{x^{2} + 12 x + 35}$ .

Factoring the denominator on the right hand side gives $LaTeX: \displaystyle \left(x + 5\right) \left(x + 7\right)$ . This gives the LCD as $LaTeX: \displaystyle \left(x + 5\right) \left(x + 7\right)$ . Multiplying by the LCD gives $LaTeX: \displaystyle x \left(x + 5\right) - x - 7 = -2$ . Getting zero on one side gives $LaTeX: \displaystyle x^{2} + 4 x - 5=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 1\right) \left(x + 5\right)=0$ . The two possible solutions are $LaTeX: \displaystyle x = 1$ and $LaTeX: \displaystyle x = -5$ . Checking the possible solutions gives:
Since $LaTeX: \displaystyle 1$ is not zero of the denominator it is a solution. Since $LaTeX: \displaystyle -5$ is zero of the denominator it is not in the domain and must be rejected as a solution.