Help me with this maths question?

1) Subtracting an odd number from an odd number always results in an even number.

The definition of an odd number is ( 2n + 1 ), so

( 2n + 3 ) - ( 2n + 1 ) = 2

2) Subtracting even numbers from even numbers also always results in an even number.

( 2n + 2 ) - ( 2n ) = 2

3) Squaring an odd number always results in an odd number.

( 2n + 1 )^2 = 4n^2 + 4n + 1 <-- not factorable by 2

4) Squaring an even number always results in an even number

( 2n )^2 = 4n^2 <-- factorable by 2

5) So if n is odd, the expression calls for squaring two even numbers.

That produces 2 even perfect squares.

Their difference is even (see statement 2)

6) If n is even, the expression calls for squaring two odd numbers.

That produces 2 odd perfect squares.

Their difference is also even (see statement 1)

Difference between (n+1)^2 and (n-1)^2 can be written as 4n

(n+1)^2 - (n-1)^2=4n

So if n is odd number, 4n will be even, and their product is always divisible by 2 and therefore the product is a even number.

Further if n is an even number, 4n will be even, and their product will still be divisible by 2 and therefore the product is a even number.