Solve $LaTeX: \displaystyle \log_{12}(x + 12)+\log_{12}(x + 5) = 2$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 17 x + 60)=2$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 17 x + 60=12^{2}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 17 x - 84=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 4\right) \left(x + 21\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -21$ and $LaTeX: \displaystyle x = 4$ . The domain of the original is $LaTeX: \displaystyle \left(-12, \infty\right) \bigcap \left(-5, \infty\right)=\left(-5, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -21$ is not a solution. $LaTeX: \displaystyle x=4$ is a solution.