To determine which given product matches the expression [tex]\(\left(4 x \sqrt{5 x^2} + 2 x^2 \sqrt{6}\right)^2\)[/tex], we first need to simplify the expression step by step.
1. Simplify Inside the Parentheses:
Start with the expression inside the parentheses:
[tex]\[
4 x \sqrt{5 x^2} + 2 x^2 \sqrt{6}
\][/tex]
2. Simplify Each Term:
- For the first term [tex]\(4 x \sqrt{5 x^2}\)[/tex]:
[tex]\[
4 x \sqrt{5 x^2} = 4 x \cdot \sqrt{5} \cdot \sqrt{x^2} = 4 x \cdot \sqrt{5} \cdot x = 4 x^2 \sqrt{5}
\][/tex]
- For the second term [tex]\(2 x^2 \sqrt{6}\)[/tex]:
[tex]\[
2 x^2 \sqrt{6}
\][/tex]
Now the expression inside the parentheses is:
[tex]\[
4 x^2 \sqrt{5} + 2 x^2 \sqrt{6}
\][/tex]
3. Square the Entire Expression:
Square the simplified expression:
[tex]\[
\left(4 x^2 \sqrt{5} + 2 x^2 \sqrt{6}\right)^2
\][/tex]
Use the binomial expansion formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex] where [tex]\(a = 4 x^2 \sqrt{5}\)[/tex] and [tex]\(b = 2 x^2 \sqrt{6}\)[/tex]:
- Compute [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = \left(4 x^2 \sqrt{5}\right)^2 = 16 x^4 \cdot 5 = 80 x^4
\][/tex]
- Compute [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = \left(2 x^2 \sqrt{6}\right)^2 = 4 x^4 \cdot 6 = 24 x^4
\][/tex]
- Compute [tex]\(2ab\)[/tex]:
[tex]\[
2ab = 2 \cdot \left(4 x^2 \sqrt{5}\right) \cdot \left(2 x^2 \sqrt{6}\right) = 2 \cdot 8 x^4 \cdot \sqrt{30} = 16 x^4 \sqrt{30}
\][/tex]
Adding all these together gives:
[tex]\[
\left(4 x^2 \sqrt{5} + 2 x^2 \sqrt{6}\right)^2 = 80 x^4 + 24 x^4 + 16 x^4 \sqrt{30}
\][/tex]
[tex]\[
= 104 x^4 + 16 x^4 \sqrt{30}
\][/tex]
4. Identify the Correct Option:
Comparing with the given options, we see that the right match is:
[tex]\[
104 x^4 + 16 x^4 \sqrt{30}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{104 x^4 + 16 x^4 \sqrt{30}}
\][/tex]
Which corresponds to option 4.
