Solve $LaTeX: \displaystyle \frac{x}{x + 4} - \frac{2}{x + 1}=- \frac{6}{x^{2} + 5 x + 4}$ .

Factoring the denominator on the right hand side gives $LaTeX: \displaystyle \left(x + 1\right) \left(x + 4\right)$ . This gives the LCD as $LaTeX: \displaystyle \left(x + 1\right) \left(x + 4\right)$ . Multiplying by the LCD gives $LaTeX: \displaystyle x \left(x + 1\right) - 2 x - 8 = -6$ . Getting zero on one side gives $LaTeX: \displaystyle x^{2} - x - 2=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 2\right) \left(x + 1\right)=0$ . The two possible solutions are $LaTeX: \displaystyle x = -1$ and $LaTeX: \displaystyle x = 2$ . Checking the possible solutions gives:
Since $LaTeX: \displaystyle 2$ is not zero of the denominator it is a solution. Since $LaTeX: \displaystyle -1$ is zero of the denominator it is not in the domain and must be rejected as a solution.