Answer:
(x (x - 4) (x - 1))/(2 (x + 4))
Step-by-step explanation:
Simplify the following:
((x^2 - 16) (x^3 - 2 x^2 + x))/((2 x + 8) (x^2 + 3 x - 4))
The factors of -4 that sum to 3 are 4 and -1. So, x^2 + 3 x - 4 = (x + 4) (x - 1):
((x^2 - 16) (x^3 - 2 x^2 + x))/((x + 4) (x - 1) (2 x + 8))
Factor 2 out of 2 x + 8:
((x^2 - 16) (x^3 - 2 x^2 + x))/(2 (x + 4) (x + 4) (x - 1))
x^2 - 16 = x^2 - 4^2:
((x^2 - 4^2) (x^3 - 2 x^2 + x))/(2 (x + 4) (x + 4) (x - 1))
Factor the difference of two squares. x^2 - 4^2 = (x - 4) (x + 4):
((x - 4) (x + 4) (x^3 - 2 x^2 + x))/(2 (x + 4) (x + 4) (x - 1))
Factor x out of x^3 - 2 x^2 + x:
(x (x^2 - 2 x + 1) (x - 4) (x + 4))/(2 (x + 4) (x + 4) (x - 1))
The factors of 1 that sum to -2 are -1 and -1. So, x^2 - 2 x + 1 = (x - 1) (x - 1):
(x (x - 1) (x - 1) (x - 4) (x + 4))/(2 (x + 4) (x + 4) (x - 1))
(x - 1) (x - 1) = (x - 1)^2:
(x (x - 1)^2 (x - 4) (x + 4))/(2 (x + 4) (x + 4) (x - 1))
((x - 4) (x + 4) x (x - 1)^2)/(2 (x + 4) (x + 4) (x - 1)) = (x + 4)/(x + 4)×((x - 4) x (x - 1)^2)/(2 (x + 4) (x - 1)) = ((x - 4) x (x - 1)^2)/(2 (x + 4) (x - 1)):
(x (x - 4) (x - 1)^2)/(2 (x + 4) (x - 1))
Cancel terms. ((x - 4) x (x - 1)^2)/(2 (x + 4) (x - 1)) = ((x - 4) x (x - 1)^(2 - 1))/(2 (x + 4)):
(x (x - 4) (x - 1)^(2 - 1))/(2 (x + 4))
2 - 1 = 1:
Answer: (x (x - 4) (x - 1))/(2 (x + 4))
