$ n(n + 3) + 2 = n^2 + 3n + 2 = n^2 + n + 2n + 2 = n(n + 1) + 2(n + 1) = (n + 1)(n + 2) $
AD:
$ \dfrac{(4 . 7 + 2)(6 . 9 + 2)...(100 . 103 + 2)}{(5 . 8 + 2)(7 . 10 + 2)...(99 . 102 + 2)} \\ = \dfrac{5 . 6 . 7 . 8 ..... 101 . 102}{6 . 7 . 8 . 9 ..... 100 . 101} \\ = 510 $