The
L.C.M of a² – 2a – 3 and a³ + a² + a + 1 is 

Consider, the first expression a² - ax - 3 Find the factors of first expression ⇒ a² – 2a - 3 = (a - 3)(a + 1) Now, consider, the second expression a³ + a² + a + 1 Find the factors of the second expression ⇒ a³ + a² + a + 1 = a² (a + 1) + (a + 1) ⇒ a³ + a² + a + 1 = (a² + 1)(a + 1) The common factors of the two expressions a² – 2a - 3 and a³ + a² + a + 1 is (a + 1) (a - 3) is extra factor in the first expression and (a² + 1) are the extra factor in the second expression. Therefore, the required L.C.M. of  a² – 2a – 3 and a³ + a² + a + 1 is (a + 1)(a - 3)(a² + 1).