Solve $LaTeX: \displaystyle \log_{ 4 }(x + 6) + \log_{ 4 }(x + 66) = 4$

Using the product rule for logarithms gives $LaTeX: \displaystyle \log_{ 4 }(\left(x + 6\right) \left(x + 66\right))$ and rewriting in exponential form gives $LaTeX: \displaystyle \left(x + 6\right) \left(x + 66\right) = 256$ expanding and setting the equation equal to zero gives $LaTeX: \displaystyle x^{2} + 72 x + 140 = 0$ . Factoring gives $LaTeX: \displaystyle \left(x + 2\right) \left(x + 70\right)=0$ . This gives two possible solutions $LaTeX: \displaystyle x=-70$ or $LaTeX: \displaystyle x=-2$ . $LaTeX: \displaystyle x=-70$ is an extraneous solution. The only soution is $LaTeX: \displaystyle x=-2$ .