To simplify the expression [tex]\((4 - i)(-3 + 7i) - 7i(8 + 2i)\)[/tex] into the form [tex]\(a + bi\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are rational numbers, follow these steps:
1. Expand the first product [tex]\((4 - i)(-3 + 7i)\)[/tex]:
[tex]\[
\begin{aligned}
(4 - i)(-3 + 7i) &= 4 \cdot (-3) + 4 \cdot (7i) + (-i) \cdot (-3) + (-i) \cdot (7i) \\
&= -12 + 28i + 3i - 7i^2 \\
&= -12 + 31i - 7(-1) \quad \text{(since } i^2 = -1) \\
&= -12 + 31i + 7 \\
&= -5 + 31i
\end{aligned}
\][/tex]
2. Expand the second product [tex]\((-7i)(8 + 2i)\)[/tex]:
[tex]\[
\begin{aligned}
(-7i)(8 + 2i) &= (-7i)(8) + (-7i)(2i) \\
&= -56i - 14i^2 \\
&= -56i - 14(-1) \quad \text{(since } i^2 = -1) \\
&= -56i + 14
\end{aligned}
\][/tex]
3. Combine the results of the two products:
[tex]\[
\begin{aligned}
(-5 + 31i) + (14 - 56i) &= -5 + 31i + 14 - 56i \\
&= (-5 + 14) + (31i - 56i) \\
&= 9 - 25i
\end{aligned}
\][/tex]
So, the simplified form of [tex]\((4 - i)(-3 + 7i) - 7i(8 + 2i)\)[/tex] is:
[tex]\[
9 - 25i
\][/tex]
Thus, in the form [tex]\(a + bi\)[/tex], we have [tex]\(a = 9\)[/tex] and [tex]\(b = -25\)[/tex].
