Answer
$x = -13.5, -5.5$
Work Step by Step
$\log_4 (-2x +5) + \log_4 (x+21.5) = 4$
$\log_4 (-2x+5)(x+21.5) = 4$
$4^{4} = (-2x+5)(x+21.5)$
$-2x(x+21.5)+5(x+21.5)= 256$
$-2x^{2} - 43x + 5x + 107.5 = 256$
$-2x^{2} - 38x + 107.5 - 256 = 0$
$-2x^{2} - 38x - 148.5 = 0$
$x = \frac{-b±\sqrt {b^{2}-4ac}}{2a}$
$x = \frac{-(-38)±\sqrt {(-38)^{2}-4(-2)(-148.5)}}{2(-2)}$
$x = \frac{38±\sqrt {1444-1188}}{-4}$
$x = \frac{38±\sqrt {256}}{-4}$
$x = \frac{38±16}{-4}$
$x = -13.5, -5.5$
Check:
When $x = -5.5$
$\log_4 (-2(-5.5) +5) + \log_4 ((-5.5)+21.5) \overset{?}{=} 4$
$\log_4 (11 +5) + \log_4 (16) \overset{?}{=} 4$
$\log_4 (16) + \log_4 (16) \overset{?}{=} 4$
$2 + 2 \overset{?}{=} 4$
$4 = 4$
When $x = -13.5$
$\log_4 (-2(-13.5) +5) + \log_4 ((-13.5)+21.5) \overset{?}{=} 4$
$\log_4 (27 +5) + \log_4 (8) \overset{?}{=} 4$
$\log_4 (32) + \log_4 (8) \overset{?}{=} 4$
$2.5 + 1.5 \overset{?}{=} 4$
$4 = 4$
