To find the complex number equivalent to the expression [tex]\((2 + i)(8 - 3i) - (23 + 34i)\)[/tex], we can tackle the problem step-by-step:
1. First, multiply the complex numbers [tex]\((2 + i)(8 - 3i)\)[/tex]:
- Use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(2 + i)(8 - 3i) = 2 \cdot 8 + 2 \cdot (-3i) + i \cdot 8 + i \cdot (-3i)
\][/tex]
- Perform the multiplications:
[tex]\[
2 \cdot 8 = 16
\][/tex]
[tex]\[
2 \cdot (-3i) = -6i
\][/tex]
[tex]\[
i \cdot 8 = 8i
\][/tex]
[tex]\[
i \cdot (-3i) = -3i^2
\][/tex]
- Remember that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
-3i^2 = -3(-1) = 3
\][/tex]
- Combine all the terms:
[tex]\[
16 - 6i + 8i + 3
\][/tex]
- Simplify this to:
[tex]\[
19 + 2i
\][/tex]
2. Next, subtract the complex number [tex]\((23 + 34i)\)[/tex] from the result:
- Subtract the real parts:
[tex]\[
19 - 23 = -4
\][/tex]
- Subtract the imaginary parts:
[tex]\[
2i - 34i = -32i
\][/tex]
3. Combine the results:
- The result is:
[tex]\[
-4 - 32i
\][/tex]
Thus, the complex number equivalent to the given expression [tex]\((2 + i)(8 - 3i) - (23 + 34i)\)[/tex] is [tex]\(-4 - 32i\)[/tex].
So the correct answer is:
C. [tex]\(-4 - 32i\)[/tex]
