Solve $LaTeX: \displaystyle \log_{12}(x + 75)+\log_{12}(x + 250) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 325 x + 18750)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 325 x + 18750=12^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 325 x - 1986=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 6\right) \left(x + 331\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -331$ and $LaTeX: \displaystyle x = 6$ . The domain of the original is $LaTeX: \displaystyle \left(-75, \infty\right) \bigcap \left(-250, \infty\right)=\left(-75, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -331$ is not a solution. $LaTeX: \displaystyle x=6$ is a solution.