Certainly! Let's simplify the given expression step-by-step:
We need to find the value of the product:
[tex]\[ 3 \sqrt{2} \big(5 \sqrt{6} - 7 \sqrt{3}\big) \][/tex]
We can distribute [tex]\( 3 \sqrt{2} \)[/tex] across the terms inside the parenthesis:
First, multiply [tex]\( 3 \sqrt{2} \)[/tex] by [tex]\( 5 \sqrt{6} \)[/tex]:
[tex]\[ 3 \sqrt{2} \cdot 5 \sqrt{6} \][/tex]
[tex]\[ = 3 \cdot 5 \cdot \sqrt{2} \cdot \sqrt{6} \][/tex]
[tex]\[ = 15 \cdot \sqrt{2 \cdot 6} \][/tex]
[tex]\[ = 15 \cdot \sqrt{12} \][/tex]
[tex]\[ = 15 \cdot \sqrt{4 \cdot 3} \][/tex]
[tex]\[ = 15 \cdot 2 \sqrt{3} \][/tex]
[tex]\[ = 30 \sqrt{3} \][/tex]
Next, multiply [tex]\( 3 \sqrt{2} \)[/tex] by [tex]\( 7 \sqrt{3} \)[/tex]:
[tex]\[ 3 \sqrt{2} \cdot 7 \sqrt{3} \][/tex]
[tex]\[ = 3 \cdot 7 \cdot \sqrt{2} \cdot \sqrt{3} \][/tex]
[tex]\[ = 21 \cdot \sqrt{2 \cdot 3} \][/tex]
[tex]\[ = 21 \cdot \sqrt{6} \][/tex]
[tex]\[ = 21 \sqrt{6} \][/tex]
Now, subtract the second product from the first product:
[tex]\[ 30 \sqrt{3} - 21 \sqrt{6} \][/tex]
So the simplified expression is:
[tex]\[ 30 \sqrt{3} - 21 \sqrt{6} \][/tex]
Upon examining the given options, the correct answer is:
[tex]\[ \text{Option (4): } 30 \sqrt{3} - 21 \sqrt{6} \][/tex]
Thus, the final result is [tex]\(\boxed{4}\)[/tex].
