I am trying to prove that if $x$ is even, then $x^2 + 4x + 2$ even.
When completing a proof by contradiction I reach this point where I need to use a fraction to get it in the form of the odd definition... is this legal for a domain of $\mathbb{Z}$? \begin{align} x^2 + 4x + 2 &= 32k^3 + 48k^2 + 24k + 6 \\ \implies\quad x^2 + 4x + 2 &= 2\bigl(16k^3 + 24k^2 + 12k + \tfrac{5}{2}\bigr) + 1 \end{align} Let $d = 16k^3 + 24k^2 + 12k + \tfrac{5}{2}$ for some $d$ in ______? I know the standard and answer is $\mathbb{Z}$, but it obviously won't work and I am stuck. Thanks for any advice.