Solve $LaTeX: \displaystyle \log_{10}(x + 22)+\log_{10}(x + 1) = 2$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{10}(x^{2} + 23 x + 22)=2$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 23 x + 22=10^{2}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 23 x - 78=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 3\right) \left(x + 26\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -26$ and $LaTeX: \displaystyle x = 3$ . The domain of the original is $LaTeX: \displaystyle \left(-22, \infty\right) \bigcap \left(-1, \infty\right)=\left(-1, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -26$ is not a solution. $LaTeX: \displaystyle x=3$ is a solution.