Let's check each of the conditions.
A)-If x is odd and xy+x is even, that means that x*(y+1) is even. For it to be even, at least one, x or y+1, must be even, and because it isn't x, it must be y+1. And, if y+1 is even, then y is certainly odd.
B)-This is not true if x=3 and y=1. 3*1+3=6 is even and the statement "If x is odd, then y is even" isn't true. And if it isn't true for one value of x and y, then the statement is false.
C)-This one is not true for x=2 and y=4.
D)-This one isn't true for x=2 and y=3.
E)-This one isn't true for x=1 and y=3.
So, we have that only A is always correct.
If you didn't understand some part of my post, feel free to ask.