Lines parallel to a given line I have to show that any line parallel to $Ax+ By + c =0$ is of the form $Ax + By + k =0$

How do I show this?

Thank you!
Examples don't show it in any case. You cannot prove that something holds for infinitely many things by doing nothing except checking finitely many examples.

What you need to do is remember that two lines are parallel if and only if they have the same slope or they are both vertical (of the form $x=k$ for some constant $k$). Then figure out what the slope of an arbitrary line $\alpha x+\beta y+\gamma=0$ would be in terms of $\alpha$, $\beta$, and $\gamma$ (careful when dividing by constants to consider what happens if the constant you are dividing by is equal to zero). Then see what this means if you want to find all lines $\alpha x+\beta y+\gamma = 0$ that are parallel to a given line $Ax+By+c=0$.
You can solve the following system of equations:

$$
\begin{cases}
Ax+By+c=0,\\
Ax+By+k=0
\end{cases}
$$

If you subtract the second line from the firs, you obtain $c-k=0$. Then, if $c\ne k$, you have no solution: the lines are parallel.