To evaluate the expression [tex]\( (-2 + 4i)(-3 + i) \)[/tex] and write the result in the form [tex]\( a + bi \)[/tex], we can follow these steps:
1. Use the distributive property (also known as the FOIL method for binomials) to expand the product:
[tex]\[
(-2 + 4i)(-3 + i) = (-2) \cdot (-3) + (-2) \cdot i + (4i) \cdot (-3) + (4i) \cdot i
\][/tex]
2. Now perform each of the multiplications:
[tex]\[
(-2) \cdot (-3) = 6
\][/tex]
[tex]\[
(-2) \cdot i = -2i
\][/tex]
[tex]\[
(4i) \cdot (-3) = -12i
\][/tex]
[tex]\[
(4i) \cdot i = 4i^2
\][/tex]
3. Recall that [tex]\( i^2 = -1 \)[/tex]. Thus, we have:
[tex]\[
4i^2 = 4(-1) = -4
\][/tex]
4. Combine all the results from the multiplications:
[tex]\[
6 - 2i - 12i - 4
\][/tex]
5. Combine the real parts and the imaginary parts:
[tex]\[
(6 - 4) + (-2i - 12i) = 2 - 14i
\][/tex]
So, the result of the expression [tex]\( (-2 + 4i)(-3 + i) \)[/tex] in the form [tex]\( a + bi \)[/tex] is [tex]\( 2 - 14i \)[/tex].
