Solve $LaTeX: \displaystyle \log_{6}(x + 25)+\log_{6}(x + 236) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{6}(x^{2} + 261 x + 5900)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 261 x + 5900=6^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 261 x - 1876=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 7\right) \left(x + 268\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -268$ and $LaTeX: \displaystyle x = 7$ . The domain of the original is $LaTeX: \displaystyle \left(-25, \infty\right) \bigcap \left(-236, \infty\right)=\left(-25, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -268$ is not a solution. $LaTeX: \displaystyle x=7$ is a solution.