Sure, let's go through the problem step-by-step.
We are asked to find the product of the complex number [tex]\(4i\)[/tex] and another complex number [tex]\(2 - 8i\)[/tex].
Given:
[tex]\[ 4i (2 - 8i) \][/tex]
To find the product, we will use the distributive property of multiplication over addition. Let's break it down:
1. Distribute [tex]\(4i\)[/tex] to both terms inside the parenthesis:
[tex]\[ 4i \cdot 2 + 4i \cdot (-8i) \][/tex]
2. Now calculate these products individually:
[tex]\[ 4i \cdot 2 = 8i \][/tex]
[tex]\[ 4i \cdot (-8i) = -32i^2 \][/tex]
3. Recall that [tex]\(i^2 = -1\)[/tex]. Using this identity:
[tex]\[ -32i^2 = -32 \cdot (-1) = 32 \][/tex]
4. Combine the real part and the imaginary part:
[tex]\[ 8i + 32 \][/tex]
5. It is common to write the real part first in complex numbers:
[tex]\[ 32 + 8i \][/tex]
So, the product of [tex]\(4i\)[/tex] and [tex]\(2 - 8i\)[/tex] is:
[tex]\[ 32 + 8i \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{32+8i} \][/tex]
So, the correct option is:
d. [tex]\( 32+8i \)[/tex]
