A polynomial is reducibble in Z[x] if and only if it is reducible in Q[x].

Begin this problem by writing,

y^2-18y-175=0 where y=x^6.

Find zeros and hence find the factors,

(y-25)(y+7)=0

Thus,

(x^3-25)(x^3+7)=0

Those two polyonomial are irreducible in Z[x].

Because,

x^3-25 has no zero (and has degree at most three).

x^3+7 has no zero (and has degree at most three).