To simplify the given expression [tex]\((-3 + 2i) \cdot (2 + i)\)[/tex], we can use the distributive property of multiplication over addition, often referred to as the FOIL method in the context of binomials. Here's a step-by-step solution:
1. Identify the terms: We have two complex numbers:
[tex]\[
(-3 + 2i) \quad \text{and} \quad (2 + i)
\][/tex]
2. Apply the distributive property (FOIL Method):
To multiply [tex]\((a + bi)(c + di)\)[/tex], we expand it as follows:
[tex]\[
(a + bi)(c + di) = ac + adi + bci + bdi^2
\][/tex]
For our specific problem, [tex]\(a = -3\)[/tex], [tex]\(b = 2\)[/tex], [tex]\(c = 2\)[/tex], and [tex]\(d = 1\)[/tex]. Substituting these values, we get:
[tex]\[
(-3 + 2i)(2 + i) = (-3) \cdot 2 + (-3) \cdot i + (2i) \cdot 2 + (2i) \cdot i
\][/tex]
3. Calculate the individual products:
[tex]\[
(-3) \cdot 2 = -6
\][/tex]
[tex]\[
(-3) \cdot i = -3i
\][/tex]
[tex]\[
(2i) \cdot 2 = 4i
\][/tex]
[tex]\[
(2i) \cdot i = 2i^2
\][/tex]
4. Combine the results:
[tex]\[
(-3 + 2i)(2 + i) = -6 - 3i + 4i + 2i^2
\][/tex]
5. Simplify using the fact that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
2i^2 = 2(-1) = -2
\][/tex]
So, we now have:
[tex]\[
-6 - 3i + 4i - 2
\][/tex]
6. Combine the real parts and the imaginary parts:
[tex]\[
\text{Real part: } -6 - 2 = -8
\][/tex]
[tex]\[
\text{Imaginary part: } -3i + 4i = i
\][/tex]
7. Write the final simplified expression:
[tex]\[
(-3 + 2i)(2 + i) = -8 + i
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
-8 + i
\][/tex]
