Let's solve the expression [tex]\((2i + 1)(5 - i)\)[/tex] step-by-step and determine which of the given options it is equivalent to.
First, expand the expression using the distributive property (also known as the FOIL method for binomials):
[tex]\[
(2i + 1)(5 - i) = (2i \cdot 5) + (2i \cdot -i) + (1 \cdot 5) + (1 \cdot -i)
\][/tex]
Now, solve each term individually:
1. [tex]\(2i \cdot 5 = 10i\)[/tex]
2. [tex]\(2i \cdot -i = -2i^2 =
3. \(1 \cdot 5 = 5\)[/tex]
4. [tex]\(1 \cdot -i = -i\)[/tex]
Next, note that [tex]\(i^2 = -1\)[/tex]. Therefore, the term [tex]\(-2i^2\)[/tex] simplifies as follows:
[tex]\[
-2i^2 = -2(-1) = 2
\][/tex]
Combining all the simplified terms:
[tex]\[
10i + 2 + 5 - i
\][/tex]
Group the real parts and the imaginary parts:
[tex]\[
(2 + 5) + (10i - i)
\][/tex]
Simplify these grouped terms:
[tex]\[
7 + 9i
\][/tex]
Thus, the expression [tex]\((2i + 1)(5 - i)\)[/tex] simplifies to [tex]\(7 + 9i\)[/tex].
So, the equivalent expression is option [tex]\( C \)[/tex]:
C. [tex]\(7 + 9i\)[/tex]
