Answer
$13\text{ cm }\times 13\text{ cm}$
Work Step by Step
Let's note by $x$ the length of the big cardboard square. The length of the box's base square is $x-2(4)=x-8$.
We determine $x$ using the volume $V$ of the box:
$$\begin{align*}
V&=100\\
(x-8)^2\cdot 4&=100\\
(x-8)^2&=25\\
x-8&=\pm 5\\
x-8=-5&\text{ or }x-8=5\\
x=3&\text{ or } x=13.
\end{align*}$$
Because the length of the box's base is $x-8$, we have $x>8$, so only the solution $x=13$ fits.
The cardboard piece is $13\text{ cm }\times 13\text{ cm}$.
