The function f(x)=(x^4-42x^2-80x+32)^3 is:

$f(x)=(x^4-42x^2-80x+32)^3$

$f'(x)=3(x^4-42x^2-80x+32)^2(4x^3-84x-80)$

$f'(x)>0$  $⇒4(x^3-21x-20)>0⇒4(x+1)(x+4)(x-5)>0$

$⇒x∈(-4,-1)∪(5,∞)$

So f (x) is monotonically increasing in $x∈(-4,-1)∪(5,∞)$