First, it's important to know that
1. Odd + odd = even ex. 1+1=2
2. 2 times any counting number is equal to even ex. 1 x 2 = 2
3. Even + odd = odd ex. 2+1=3
Then we can say that m^2 is odd because if it is added to 1, the result is even (as shown in number 1)
And because m^2 is odd, then m is odd because it sqaure root will not result to even when the square is odd.
Therefore we can say that 2n+m is odd because 2n is even (example #2) and adding odd to that will result in an odd number (#3 example).