discrete mathematics - Proof by induction or contradiction that $(4k + 3) ^2 - (4k + 3)$ is not divisible by $4$?

I have to prove that $(4k + 3) ^2 - (4k + 3)$ is not divisible by $4$.

What would be the best approach for this, proof by induction or contradiction?

I've tried both and haven't got very far. Any hints would be appreciated, I'm not looking for a full answer..I wanna try it out myself but I need some help on where to begin.

Answer

The polynomial is simple enough that it’s no problem simply to multiply it out:

$$\begin{align*}
(4k+3)^2-(4k+3)&=16k^2+24k+9-4k-3\\
&=16k^2+20k+6\\
&=4\left(4k^2+5k+1\right)+2\;,

\end{align*}$$

which is clearly not a multiple of $4$. This is perhaps a little less elegant than njguliyev’s solution, but it works just fine.