Let's simplify the expression [tex]\((3 - 5i)(-4 + 2i)\)[/tex] step-by-step.
First, we apply the distributive property, also known as the FOIL method (First, Outer, Inner, Last):
[tex]\[
(3 - 5i)(-4 + 2i)
\][/tex]
1. First terms: Multiply the first terms of each binomial:
[tex]\[
3 \cdot (-4) = -12
\][/tex]
2. Outer terms: Multiply the outer terms of the binomials:
[tex]\[
3 \cdot 2i = 6i
\][/tex]
3. Inner terms: Multiply the inner terms of the binomials:
[tex]\[
-5i \cdot (-4) = 20i
\][/tex]
4. Last terms: Multiply the last terms of each binomial:
[tex]\[
-5i \cdot 2i = -10i^2
\][/tex]
Remember, [tex]\(i^2 = -1\)[/tex]. So, we substitute [tex]\(i^2\)[/tex] with [tex]\(-1\)[/tex]:
[tex]\[
-10i^2 = -10(-1) = 10
\][/tex]
Now, we combine all these results:
[tex]\[
(3 - 5i)(-4 + 2i) = -12 + 6i + 20i + 10
\][/tex]
Combine the like terms (real parts and imaginary parts):
Real part:
[tex]\[
-12 + 10 = -2
\][/tex]
Imaginary part:
[tex]\[
6i + 20i = 26i
\][/tex]
So, the simplified expression is:
[tex]\[
-2 + 26i
\][/tex]
Thus, the correct answer is:
A. [tex]\(-2 + 26i\)[/tex]
