<s>[INST] $n^2+3n+2$ is even for any integer $n$ I am trying to prove the following:
If n is an integer, then $n^{2}+3n+2$ is even.
I tried proving this by contradiction, assuming that $n_{2}+3n+2=2k+1$, but I couldn't continue from that point.
Thanks!
 [/INST] If $n$ is even it can be written as $n=2k$
$$\begin{array}{rcl}n^2 + 3n + 2 &=& (2k)^2 + 3(2k) + 2 \\&=& 2\cdot(2k^2+3k+1)\end{array}$$
If $n$ is odd it can be written as $n=2k+1$
$$\begin{array}{rcl}n^2 + 3n + 2 &=& (2k+1)^2 + 3(2k+1) + 2\\ &=& 4k^2+4k+1+6k+3+2 \\&=& 2\cdot(2k^2+5k+3)\end{array}$$
Hence $n^2 + 3n + 2$ is even for every integer $n$
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