Solve $LaTeX: \displaystyle \log_{6}(x + 2)+\log_{6}(x + 7) = 2$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{6}(x^{2} + 9 x + 14)=2$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 9 x + 14=6^{2}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 9 x - 22=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 2\right) \left(x + 11\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -11$ and $LaTeX: \displaystyle x = 2$ . The domain of the original is $LaTeX: \displaystyle \left(-2, \infty\right) \bigcap \left(-7, \infty\right)=\left(-2, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -11$ is not a solution. $LaTeX: \displaystyle x=2$ is a solution.