Sure, let's find the product of the expression [tex]\((x \sqrt{7} - 3 \sqrt{8}) (x \sqrt{7} - 3 \sqrt{8})\)[/tex].
To do this, we will use the formula for expanding a binomial squared: [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. In our case, [tex]\(a = x \sqrt{7}\)[/tex] and [tex]\(b = 3 \sqrt{8}\)[/tex].
First, let's identify and compute the three terms in the expanded form individually.
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = (x \sqrt{7})^2 = x^2 \cdot (\sqrt{7})^2 = x^2 \cdot 7 = 7x^2
\][/tex]
2. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (3 \sqrt{8})^2 = 9 \cdot (\sqrt{8})^2 = 9 \cdot 8 = 72
\][/tex]
3. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
2ab = 2 \cdot (x \sqrt{7}) \cdot (3 \sqrt{8}) = 2 \cdot x \cdot \sqrt{7} \cdot 3 \cdot \sqrt{8} = 6 \cdot x \cdot (\sqrt{7} \cdot \sqrt{8})
\][/tex]
We know that [tex]\(\sqrt{7} \cdot \sqrt{8} = \sqrt{56} = \sqrt{4 \cdot 14} = 2 \sqrt{14}\)[/tex]. So:
[tex]\[
2ab = 6 \cdot x \cdot 2 \sqrt{14} = 12x \sqrt{14}
\][/tex]
Thus, [tex]\(-2ab = -12x \sqrt{14}\)[/tex].
Now, combine these results to form the expanded product:
[tex]\[
(x \sqrt{7} - 3 \sqrt{8})^2 = a^2 - 2ab + b^2 = 7x^2 - 12x \sqrt{14} + 72
\][/tex]
So, the product is:
[tex]\[
7x^2 - 12x \sqrt{14} + 72
\][/tex]
Out of the given options, this corresponds to:
[tex]\[
7 x^2-12 x \sqrt{14}+72
\][/tex]
Therefore, the correct answer is:
[tex]\[
7 x^2 - 12 x \sqrt{14} + 72
\][/tex]
