Sure, let's solve the problem step-by-step.
You are given two complex numbers [tex]\((-4 - 5i)\)[/tex] and [tex]\((1 - i)\)[/tex] and you need to multiply them.
Step-by-step multiplication of two complex numbers involves distributing each term in the first complex number by each term in the second complex number.
1. Expression Setup:
[tex]\[
(-4 - 5i) \cdot (1 - i)
\][/tex]
2. Distribute the terms:
[tex]\[
((-4) \cdot (1)) + ((-4) \cdot (-i)) + ((-5i) \cdot (1)) + ((-5i) \cdot (-i))
\][/tex]
3. Perform each multiplication:
- [tex]\( (-4) \cdot (1) = -4 \)[/tex]
- [tex]\( (-4) \cdot (-i) = 4i \)[/tex]
- [tex]\( (-5i) \cdot (1) = -5i \)[/tex]
- [tex]\( (-5i) \cdot (-i) = 5i^2 \)[/tex]
4. Combine the results:
[tex]\[
-4 + 4i - 5i + 5i^2
\][/tex]
5. Recall that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
5i^2 = 5(-1) = -5
\][/tex]
6. Substitute [tex]\(5i^2\)[/tex] with [tex]\(-5\)[/tex] and combine all the terms:
[tex]\[
-4 + 4i - 5i - 5
\][/tex]
7. Combine the real parts and the imaginary parts:
- The real parts: [tex]\(-4 - 5 = -9\)[/tex]
- The imaginary parts: [tex]\(4i - 5i = -1i\)[/tex]
8. Result:
[tex]\[
-9 - 1i
\][/tex]
So, the product of [tex]\((-4 - 5i)\)[/tex] and [tex]\((1 - i)\)[/tex] is [tex]\((-9 - 1i)\)[/tex].
To summarize:
- Real part: [tex]\(-9\)[/tex]
- Imaginary part: [tex]\(-1\)[/tex]
- Resulting complex number: [tex]\((-9 - 1i)\)[/tex]
