Answer
See below
Work Step by Step
Given $A=\begin{bmatrix}
1 &2 & 3 & 4\\ 5 & 6 & 7 & 8
\end{bmatrix}$
Since $x,y,z,w \in nullspace (A)$ we obtain
$Ax=0\\
\begin{bmatrix}
1 &2 & 3 & 4\\ 5 & 6 & 7 & 8
\end{bmatrix}\begin{bmatrix}
x \\ y \\ z \\ w
\end{bmatrix}=0\\
\begin{bmatrix}
x +2y + 3z + 4w\\ 5x+ 6y+ 7z + 8w
\end{bmatrix}=0\\
\rightarrow x+2y+3z+4w=0\\
5x+6y+7z+8w=0\\
\rightarrow x=-2y-3z-4w$
Substitute $5(-2y-3z-4w)+6y+7z+8w=0\\
\rightarrow y=-2z-3w$
Then $x=-2(-2z-3w)-3z-4w\\
x=z+2w$
Hence, nullspace $(A) =\{(x,y,z,w)\in R^4:x=z+w,y=-2z-3w\}$
