Definition NSM tells us that the null space of $A$ is the solution set to the homogeneous system $\homosystem{A}$. The augmented matrix of this system is
\begin{equation*}
\begin{bmatrix}
2 & 4 & 1 & 3 & 8 & 0 \\
-1 & -2 & -1 & -1 & 1 & 0 \\
2 & 4 & 0 & -3 & 4 & 0 \\
2 & 4 & -1 & -7 & 4 & 0
\end{bmatrix}
\end{equation*}
To solve the system, we row-reduce the augmented matrix and obtain,
\begin{equation*}
\begin{bmatrix}
\leading{1} & 2 & 0 & 0 & 5 & 0 \\
0 & 0 & \leading{1} & 0 & -8 & 0 \\
0 & 0 & 0 & \leading{1} & 2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\end{equation*}
This matrix represents a system with equations having three dependent variables ($x_1$, $x_3$, and $x_4$) and two independent variables ($x_2$ and $x_5$). These equations rearrange to
\begin{align*}
x_1&=-2x_2-5x_5
&
x_3&=8x_5
&
x_4&=-2x_5
\end{align*}
So we can write the solution set (which is the requested null space) as
\begin{equation*}
\nsp{A}=\setparts{
\colvector{-2x_2-5x_5\\x_2\\8x_5\\-2x_5\\x_5}
}{
x_2, x_5\in\complex{\null}
}
\end{equation*}