**Prove that for every integer x, $\displaystyle x^2 + x $ is even.**

This is what i have so far:

Prove by contradiction,

Suppose $\displaystyle x^2 + x $ is ODD

$\displaystyle x = 2k + 1 $ (where k is any integer)

now plug in (2k + 1) for $\displaystyle x^2 + x $ is ODD

$\displaystyle (2k + 1)^2 + (2k + 1) $ = ODD

$\displaystyle 4k^2 + 4k + 1 + 2k + 1 $ = ODD

$\displaystyle 4k^2 + 6k + 2 $ = ODD

This is False!, ex. when k = 1, $\displaystyle 4k^2 + 6k + 2 $ is EVEN

Therefore $\displaystyle x^2 + x $ is EVEN

Does this all seem correct? i am very new to proofs. any criticism is appreciated

Thank you