To simplify the expression [tex]\((2 \sqrt{7} - 4)(3 \sqrt{7} - 1)\)[/tex], we'll follow a systematic approach. Let's break it down step by step.
First, distribute each term in the first parenthesis by each term in the second parenthesis (using the distributive property, also known as the FOIL method: First, Outer, Inner, Last).
1. First Terms:
[tex]\[
(2 \sqrt{7})(3 \sqrt{7})
\][/tex]
When multiplied, this becomes:
[tex]\[
2 \cdot 3 \cdot \sqrt{7} \cdot \sqrt{7} = 6 \cdot 7 = 42
\][/tex]
2. Outer Terms:
[tex]\[
(2 \sqrt{7})(-1)
\][/tex]
When multiplied, this becomes:
[tex]\[
2 \sqrt{7} \cdot (-1) = -2 \sqrt{7}
\][/tex]
3. Inner Terms:
[tex]\[
(-4)(3 \sqrt{7})
\][/tex]
When multiplied, this becomes:
[tex]\[
-4 \cdot 3 \cdot \sqrt{7} = -12 \sqrt{7}
\][/tex]
4. Last Terms:
[tex]\[
(-4)(-1)
\][/tex]
When multiplied, this becomes:
[tex]\[
-4 \cdot (-1) = 4
\][/tex]
Now, combine all these results:
[tex]\[
42 + (-2\sqrt{7}) + (-12\sqrt{7}) + 4
\][/tex]
Next, combine like terms. Here, we have two types of terms: constant terms and terms involving [tex]\(\sqrt{7}\)[/tex].
1. Combine the constant terms:
[tex]\[
42 + 4 = 46
\][/tex]
2. Combine the terms with [tex]\(\sqrt{7}\)[/tex]:
[tex]\[
-2 \sqrt{7} - 12 \sqrt{7} = -14 \sqrt{7}
\][/tex]
Finally, the simplified expression is:
[tex]\[
\boxed{46 - 14\sqrt{7}}
\][/tex]
