Is the positive integer m odd?
I. $$m^2 + 2m$$ is even
II. $$m^2 + m$$ is even

I.

let put $$m=1 and m=2$$

when $$m=1$$ ,$$m^2+2m=1^2+2=3.$$ (which is an odd number.)

when$$m=2$$ ,$$m^2+2m=2^2+4=8.$$(which is an even number.)

So we can say that $$m^2+2m$$ is only even when m is an even number.

II.

let put $$m=1 and m=2$$

when $$m=1$$ ,$$m^2+m=1^2+1=2.$$

when $$m=2$$ ,$$m^2+m=2^2+2=6.$$

both the numbers are even numbers. So we can not say that whether m is odd or even.

So, A is correct choice.