Induction proof verification?

Using induction, I have to prove that $n^2+n$ is divisible by 2.
Here's how I did it, and I wanted to know if this is considered valid. I started with $(k+1)^2+(k+1)$, and after simplifying and factoring, I got $(k+2)(k+1)$. From this, can't I just say that if $k$ is odd, then the term $(k+1)$ becomes an even number, and when an even number is multiplied by an odd number, you get an even number which is always divisible by $2$. Then, the same logic can be applied for when $k$ is even and added to the term $(k+2)$? $Q.E.D$...?