To find the product of the given expression \( (x \sqrt{7} - 3 \sqrt{8})(x \sqrt{7} - 3 \sqrt{8}) \), let's follow the steps for expanding this product.
1. Rewrite the expression using the distributive property (FOIL method):
[tex]\[
(x \sqrt{7} - 3 \sqrt{8})(x \sqrt{7} - 3 \sqrt{8})
\][/tex]
2. Expand the expression:
[tex]\[
= (x \sqrt{7}) (x \sqrt{7}) + (x \sqrt{7}) (-3 \sqrt{8}) + (-3 \sqrt{8}) (x \sqrt{7}) + (-3 \sqrt{8}) (-3 \sqrt{8})
\][/tex]
3. Simplify each term individually:
- [tex]\[
(x \sqrt{7}) (x \sqrt{7}) = x^2 \cdot 7 = 7x^2
\][/tex]
- [tex]\[
(x \sqrt{7}) (-3 \sqrt{8}) = -3x \cdot \sqrt{7 \cdot 8} = -3x \sqrt{56}
\][/tex]
- [tex]\[
(-3 \sqrt{8}) (x \sqrt{7}) = -3x \sqrt{7 \cdot 8} = -3x \sqrt{56}
\][/tex]
- [tex]\[
(-3 \sqrt{8}) (-3 \sqrt{8}) = 9 \cdot 8 = 72
\][/tex]
4. Combine the terms:
[tex]\[
7x^2 + (-3x \sqrt{56}) + (-3x \sqrt{56}) + 72
\][/tex]
5. Combine like terms together:
- The middle terms can be combined:
[tex]\[
-3x \sqrt{56} - 3x \sqrt{56} = -6x \sqrt{56}
\][/tex]
Note that \(\sqrt{56} = \sqrt{4 \cdot 14} = 2\sqrt{14} \):
[tex]\[
-6x \sqrt{56} = -6x \cdot 2\sqrt{14} = -12x \sqrt{14}
\][/tex]
6. Put everything together:
[tex]\[
7x^2 - 12x \sqrt{14} + 72
\][/tex]
So, the result is \(7x^2 - 12x \sqrt{14} + 72\).
The correct answer is:
[tex]\[
7 x^2 - 12 x \sqrt{14} + 72
\][/tex]
