Answer
(2, 0) (-3/2,0)
See graph below.
Work Step by Step
$P(x) = 0.125 (2x^4 + 3x^3 - 16x -24)^2$
$P(x) = \frac {1}{8} (x^3 (2x +3) - 8 (2x + 3))^2$
$P(x) = \frac {1}{8} ((x^3 - 8) (2x + 3))^2$
$P(x) = \frac {1}{8} (x-2)^2(x^2 + 2x + 4)^2 (2x+3)^2$
Thus the zeros are (2, 0) (-3/2,0)