Certainly! Let's work through the problem step-by-step to find the product of the complex numbers [tex]\((-4 + 4i)\)[/tex] and [tex]\((3 + 2i)\)[/tex].
### Step 1: Recall the formula for multiplying complex numbers
For any two complex numbers [tex]\( (a + bi) \)[/tex] and [tex]\( (c + di) \)[/tex], their product is given by:
[tex]\[
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
\][/tex]
### Step 2: Identify the real and imaginary parts
For the given complex numbers:
[tex]\[
(-4 + 4i) \quad \text{and} \quad (3 + 2i)
\][/tex]
We have:
- [tex]\( a = -4 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 3 \)[/tex]
- [tex]\( d = 2 \)[/tex]
### Step 3: Multiply using the formula
Using the formula, we find the real part and the imaginary part separately.
#### Real Part:
The real part is given by:
[tex]\[
ac - bd
\][/tex]
Substituting the values:
[tex]\[
(-4) \cdot 3 - 4 \cdot 2 = -12 - 8 = -20
\][/tex]
#### Imaginary Part:
The imaginary part is given by:
[tex]\[
ad + bc
\][/tex]
Substituting the values:
[tex]\[
(-4) \cdot 2 + 4 \cdot 3 = -8 + 12 = 4
\][/tex]
### Step 4: Combine the real and imaginary parts
Combining the real part and the imaginary part, we get:
[tex]\[
(-4 + 4i) \cdot (3 + 2i) = -20 + 4i
\][/tex]
### Conclusion
So, the product of the complex numbers [tex]\((-4 + 4i)\)[/tex] and [tex]\((3 + 2i)\)[/tex] is:
[tex]\[
\boxed{-20 + 4i}
\][/tex]
