Given that [tex]\( i = \sqrt{-1} \)[/tex], which of the following is equal to [tex]\( (11 + 4i)(2 - 5i) \)[/tex]?

A. 2
B. 42
C. [tex]\( 2 - 75i \)[/tex]
D. [tex]\( 22 - 20i \)[/tex]
E. [tex]\( 42 - 47i \)[/tex]



Answer :

To solve for [tex]\((11 + 4i)(2 - 5i)\)[/tex], let's follow these steps:

1. Distribute each term in the first binomial by each term in the second binomial:
[tex]\[ (11 + 4i)(2 - 5i) = 11 \cdot 2 + 11 \cdot (-5i) + 4i \cdot 2 + 4i \cdot (-5i) \][/tex]

2. Compute each of these products:
[tex]\[ 11 \cdot 2 = 22 \][/tex]
[tex]\[ 11 \cdot (-5i) = -55i \][/tex]
[tex]\[ 4i \cdot 2 = 8i \][/tex]
[tex]\[ 4i \cdot (-5i) = -20i^2 \][/tex]

3. Simplify the product of imaginary units [tex]\(i^2\)[/tex]:
Since [tex]\(i^2 = -1\)[/tex], we have:
[tex]\[ -20i^2 = -20(-1) = 20 \][/tex]

4. Add up all the terms, combining like terms:
[tex]\[ 22 + (-55i) + 8i + 20 \][/tex]

5. Combine the real and imaginary parts separately:
The real parts:
[tex]\[ 22 + 20 = 42 \][/tex]
The imaginary parts:
[tex]\[ -55i + 8i = -47i \][/tex]

Therefore, the expression [tex]\((11 + 4i)(2 - 5i)\)[/tex] simplifies to:
[tex]\[ 42 - 47i \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{42 - 47i} \][/tex]

From the given options, the correct match is:
[tex]\[ \boxed{E} \][/tex]

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