If $P(x) = (x^3 - 8) ( x +1) \, and \ Q(x) = (x^3 +1) (x-2),$ the LCM of P(x) and Q(x) is :

$(x+1)(x-1)(x^2 +2x+4)(x^2 -x+1)$

We know that,

(a - b)³ = a³ - b³ - 3ab(a-b)

(a + b)³ = a³ + b³ + 3ab(a+b)

Given,

P(x) = (x³ - 8)(x + 1)

= P(x) = (x - 2)(x² + 2x + 4)(x + 1) 

 Q(x) = (x³ + 1)(x - 2)

= Q(x) = (x + 1)(x² - x + 1(x - 2)

Now,

The LCM of P(x) and Q(x) is = (x + 1)(x - 2)(x² + 2x + 4)(x² - x + 1)