Let's solve the problem step-by-step.
We are given the expression:
[tex]\[
\sqrt{4 x^2} \cdot \sqrt{20 x^2}
\][/tex]
First, we need to use the property of square roots that states:
[tex]\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\][/tex]
Applying this property to our expression:
[tex]\[
\sqrt{4 x^2} \cdot \sqrt{20 x^2} = \sqrt{(4 x^2) \cdot (20 x^2)}
\][/tex]
Next, we multiply the terms inside the square root:
[tex]\[
(4 x^2) \cdot (20 x^2) = 4 \cdot 20 \cdot x^2 \cdot x^2 = 80 x^4
\][/tex]
Hence, the expression now becomes:
[tex]\[
\sqrt{80 x^4}
\][/tex]
Now, we simplify the square root. Recall that:
[tex]\[
\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}
\][/tex]
In our case:
[tex]\[
\sqrt{80 x^4} = \sqrt{80} \cdot \sqrt{x^4}
\][/tex]
We know that:
[tex]\[
\sqrt{x^4} = (x^4)^{1/2} = x^2
\][/tex]
Thus, the expression simplifies to:
[tex]\[
\sqrt{80} \cdot x^2
\][/tex]
Consequently:
[tex]\[
\sqrt{80 x^4} = \sqrt{80} \cdot x^2
\][/tex]
We are left with:
[tex]\[
\sqrt{80 x^2}
\][/tex]
Among the given choices, the equivalent expression is:
C. [tex]\(\sqrt{80 x^2}\)[/tex]
Therefore, the correct choice is [tex]\( \boxed{\text{C}} \)[/tex].
