Answer
$x = -7$
Work Step by Step
$\log_5 (x+8) + \log_5 (x+12) = 1$
$\log_5 (x+8)(x+12) = 1$
$5^{1} = (x+8)(x+12)$
$5 = (x+8)(x+12)$
$x(x+12)+8(x+12) = 5$
$x^{2} + 12x + 8x + 96 = 5$
$x^{2} + 20x + 96 - 5 = 0$
$x^{2} + 20x + 91 = 0$
$x^{2} + 7x + 13x + 91 = 0$
$x(x+7) + 13(x+7) = 0$
$(x+13)(x+7) = 0$
$x = -13, -7$
Check:
$\log_5 (-13+8) + \log_5 (-13+12) \overset{?}{=} 1$
$\log_5 (-5) + \log_5 (-1) \overset{?}{=} 1$
Since $\log_5$ is undefined for negative numbers, $x=-13$ is not a solution.
$\log_5 (-7+8) + \log_5 (-7+12) \overset{?}{=} 1$
$\log_5 (1) + \log_5 (5) \overset{?}{=} 1$
$\log_5 (1)(5) \overset{?}{=} 1$
$\log_5 (5) \overset{?}{=} 1$
$1 = 1$