To solve the problem, we start with the given expression and simplify it step-by-step.
We are given the radical expression:
[tex]\[
\sqrt{2 x} \cdot \sqrt{x + 3}
\][/tex]
Step 1: Combine the radicals under a single square root. The property of radicals we will use is:
[tex]\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\][/tex]
Using this property, we combine the terms under one square root:
[tex]\[
\sqrt{2 x} \cdot \sqrt{x + 3} = \sqrt{(2 x) \cdot (x + 3)}
\][/tex]
Step 2: Simplify the expression inside the square root. We use distributive property to multiply the terms inside the square root:
[tex]\[
(2 x) \cdot (x + 3) = 2 x \cdot x + 2 x \cdot 3
\][/tex]
[tex]\[
= 2 x^2 + 6 x
\][/tex]
Therefore, our expression under a single square root becomes:
[tex]\[
\sqrt{2 x} \cdot \sqrt{x + 3} = \sqrt{2 x^2 + 6 x}
\][/tex]
Step 3: Match our simplified expression with the given choices. We find that option A matches our simplified expression:
Option A: [tex]\(\sqrt{2 x^2 + 6 x}\)[/tex]
Thus, the equivalent expression is:
Answer: A. [tex]\(\sqrt{2 x^2+6 x}\)[/tex]
