Solve $LaTeX: \displaystyle \frac{x}{x + 6} - \frac{4}{x + 8}=\frac{8}{x^{2} + 14 x + 48}$ .

Factoring the denominator on the right hand side gives $LaTeX: \displaystyle \left(x + 6\right) \left(x + 8\right)$ . This gives the LCD as $LaTeX: \displaystyle \left(x + 6\right) \left(x + 8\right)$ . Multiplying by the LCD gives $LaTeX: \displaystyle x \left(x + 8\right) - 4 x - 24 = 8$ . Getting zero on one side gives $LaTeX: \displaystyle x^{2} + 4 x - 32=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 4\right) \left(x + 8\right)=0$ . The two possible solutions are $LaTeX: \displaystyle x = 4$ and $LaTeX: \displaystyle x = -8$ . Checking the possible solutions gives:
Since $LaTeX: \displaystyle -8$ is zero of the denominator it is not in the domain and must be rejected as a solution. Since $LaTeX: \displaystyle 4$ is not zero of the denominator it is a solution.