Answer
$c = -7$
Work Step by Step
$\log_5 (c + 12) + \log_5 (c+8) - 2 = -1$
$\log_5 (c+12)(c+8) = -1 + 2$
$\log_5 (c+12)(c+8) = 1$
$5^{1} = (c+12)(c+8)$
$5= (c+12)(c+8)$
$5= c(c+8)+12(c+8)$
$5 = c^{2} + 8c+12c+96$
$c^{2} + 20c + 96 - 5 = 0$
$c^{2} + 20c + 91 = 0$
$c = \frac{-(20)±\sqrt {(20)^{2}-4(1)(91)}}{2(1)}$
$c = \frac{-20±\sqrt {400-4(1)(91)}}{2}$
$c = \frac{-20±\sqrt {400-364}}{2}$
$c = \frac{-20±\sqrt {36}}{2}$
$c = \frac{-20±6}{2}$
$c = -7, -13$
Since we can't take the log of a negative number, the only possible solution is $c = -7$.
Check:
$\log_5 (-7+ 12) + \log_5 ((-7)+8) - 2 \overset{?}{=} -1$
$\log_5 (5) + \log_5 (1) - 2 \overset{?}{=} -1$
$1(1) + 0 - 2 \overset{?}{=} -1$
$1 - 2 \overset{?}{=} -1$
$-1 = -1$
