To solve for the difference of the complex numbers [tex]\( (5 + 9i) \)[/tex] and [tex]\( (-4 - 2i) \)[/tex], follow these steps:
1. Understand the problem:
We need to find the difference of two complex numbers. Recall that a complex number has a real part and an imaginary part, and they are generally written in the form [tex]\( a + bi \)[/tex], where [tex]\( a \)[/tex] is the real part, and [tex]\( bi \)[/tex] is the imaginary part.
2. Set up the expression:
We are given the complex numbers [tex]\( (5 + 9i) \)[/tex] and [tex]\( (-4 - 2i) \)[/tex] and asked to find their difference:
[tex]\[
(5 + 9i) - (-4 - 2i)
\][/tex]
3. Distribute the negative sign:
When subtracting complex numbers, distribute the negative sign across the second complex number:
[tex]\[
(5 + 9i) - (-4 - 2i) = (5 + 9i) + (4 + 2i)
\][/tex]
4. Combine like terms:
Add the real parts together and the imaginary parts together:
[tex]\[
\text{Real part:} \quad 5 + 4 = 9
\][/tex]
[tex]\[
\text{Imaginary part:} \quad 9i + 2i = 11i
\][/tex]
5. Write the result:
Combine the results from step 4 to form the difference:
[tex]\[
9 + 11i
\][/tex]
Thus, the difference of the complex numbers [tex]\( (5 + 9i) \)[/tex] and [tex]\( (-4 - 2i) \)[/tex] is [tex]\( 9 + 11i \)[/tex].
The correct answer is:
[tex]\[
\boxed{9 + 11i}
\][/tex]
Therefore, the answer aligns with option D.
