(a)
Use the fact that [tex]a^2-b^2=(a+b)(a-b)[/tex]. In this case a=n+1, b=n-1.
I am using "4|expression" to denote that 4 should divide the expression:
[tex]4|[(n+1)^2-(n-1)^2]\implies 4|(n+1+n-1)(n+1-n+1)\implies\\\implies4|2n\cdot2\implies 4|4n \,\,\,\mbox{true for all }n\in N[/tex]
(b)
Same trick
[tex]8|[(2n+3)^2-(2n-1)^2]\implies 8|(2n+3+2n-1)(2n+3-2n+1)\implies\\\implies 8|(2n+2)\cdot4 \implies 8|8(n+1)\,\,\,\mbox{true for all } n\in N[/tex]
