To simplify the given expression [tex]\((8 + 2i) - (6 - 3i) + (4 - i)\)[/tex], we first need to deal with the subtraction and addition of complex numbers step-by-step.
1. Start with the given expression:
[tex]\[
(8 + 2i) - (6 - 3i) + (4 - i)
\][/tex]
2. Distribute the subtraction across the second complex number:
[tex]\[
(8 + 2i) - 6 + 3i + (4 - i)
\][/tex]
3. Combine like terms:
[tex]\[
8 - 6 + 4 + 2i + 3i - i
\][/tex]
4. Simplify the real parts:
[tex]\[
(8 - 6 + 4) + (2i + 3i - i)
\][/tex]
Breaking it down for clarity:
- Real part: [tex]\( 8 - 6 + 4 = 6 \)[/tex]
- Imaginary part: [tex]\( 2i + 3i - i = 4i \)[/tex]
5. Combine the simplified real and imaginary parts:
[tex]\[
6 + 4i
\][/tex]
Therefore, the simplified form of [tex]\((8 + 2i) - (6 - 3i) + (4 - i)\)[/tex] is [tex]\(\boxed{6 + 4i}\)[/tex]. So the correct answer is:
[tex]\[
\boxed{6 + 4i}
\][/tex]
