To simplify the expression [tex]\((6 + 4\sqrt{3})(2 - 3\sqrt{3})\)[/tex], we will use the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last). Here are the steps:
1. Multiply the First terms:
[tex]\[
6 \times 2 = 12
\][/tex]
2. Multiply the Outer terms:
[tex]\[
6 \times (-3\sqrt{3}) = -18\sqrt{3}
\][/tex]
3. Multiply the Inner terms:
[tex]\[
4\sqrt{3} \times 2 = 8\sqrt{3}
\][/tex]
4. Multiply the Last terms:
[tex]\[
4\sqrt{3} \times (-3\sqrt{3})
\][/tex]
Simplify the multiplication within the square root:
[tex]\[
4 \times -3 \times (\sqrt{3} \times \sqrt{3}) = 4 \times -3 \times 3 = -36
\][/tex]
Next, we combine all these products:
- Constant terms:
[tex]\[
12 - 36 = -24
\][/tex]
- Square root terms:
[tex]\[
-18\sqrt{3} + 8\sqrt{3} = -10\sqrt{3}
\][/tex]
Therefore, combining the constant term and the square root term, the simplified expression is:
[tex]\[
\boxed{-24 - 10\sqrt{3}}
\][/tex]
