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------------------------------------------------ Multiply and simplify the product: [tex]\((8 - 5i)^2\)[/tex]

Select the product:

A. 39

B. 89

C. 39 - 80i

D. 89 - 80i

To solve the problem [tex]\((8-5i)^2\)[/tex], we will use the formula for squaring a binomial [tex]\((a-b)^2 = a^2 - 2ab + b^2\)[/tex].

Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].

1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 8^2 = 64
\][/tex]

2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2ab = -2 \cdot 8 \cdot 5i = -80i
\][/tex]

3. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (5i)^2 = 25i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], we have:
[tex]\[
25i^2 = 25(-1) = -25
\][/tex]

4. Combine all the parts together:
[tex]\[
(8-5i)^2 = a^2 - 2ab + b^2 = 64 - 80i - 25
\][/tex]

5. Simplify the expression:
[tex]\[
64 - 25 = 39
\][/tex]
Hence, the simplified product is:
[tex]\[
39 - 80i
\][/tex]

Therefore, the product is [tex]\(39 - 80i\)[/tex].