To solve the expression [tex]\((3 \sqrt{3} - 3i)(-4i)\)[/tex], we should follow these steps:
1. Distribute [tex]\(-4i\)[/tex] through the binomial [tex]\(3 \sqrt{3} - 3i\)[/tex]:
[tex]\[
(3 \sqrt{3} - 3i)(-4i) = 3 \sqrt{3} \cdot (-4i) + (-3i) \cdot (-4i)
\][/tex]
2. Calculate each individual product:
[tex]\[
3 \sqrt{3} \cdot (-4i) = -12i \sqrt{3}
\][/tex]
[tex]\[
(-3i) \cdot (-4i) = 12i^2
\][/tex]
Recall that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
12i^2 = 12(-1) = -12
\][/tex]
3. Combine the results:
[tex]\[
-12i \sqrt{3} + (-12) = -12 - 12i \sqrt{3}
\][/tex]
So, the expression [tex]\((3 \sqrt{3} - 3i)(-4i)\)[/tex] simplifies to:
[tex]\[
-12 - 12i \sqrt{3}
\][/tex]
Therefore, the final answer is:
[tex]\[
-12 - 12i \sqrt{3}
\][/tex]
