To simplify the given expression [tex]\((3+7i)(-2-2i) - 4i(5-9i)\)[/tex] into the form [tex]\(a+bi\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are rational numbers, let's break it down step-by-step.
### Simplifying [tex]\((3 + 7i)(-2 - 2i)\)[/tex]:
First, we distribute each term in the first expression:
[tex]\[
(3 + 7i)(-2 - 2i) = 3(-2) + 3(-2i) + 7i(-2) + 7i(-2i)
\][/tex]
Calculating each part separately:
[tex]\[
3(-2) = -6
\][/tex]
[tex]\[
3(-2i) = -6i
\][/tex]
[tex]\[
7i(-2) = -14i
\][/tex]
[tex]\[
7i(-2i) = -14i^{2}
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], we have:
[tex]\[
7i(-2i) = -14(-1) = 14
\][/tex]
Combining these results:
[tex]\[
(3 + 7i)(-2 - 2i) = -6 - 6i - 14i + 14
\][/tex]
Grouping the real and imaginary parts together:
[tex]\[
-6 + 14 + (-6i - 14i) = 8 - 20i
\][/tex]
### Simplifying [tex]\(-4i(5 - 9i)\)[/tex]:
Next, we distribute each term in the second expression:
[tex]\[
-4i(5 - 9i) = -4i(5) + (-4i)(-9i)
\][/tex]
Calculating each part separately:
[tex]\[
-4i(5) = -20i
\][/tex]
[tex]\[
-4i(-9i) = 36i^{2}
\][/tex]
Again, using [tex]\(i^2 = -1\)[/tex]:
[tex]\[
36i^{2} = 36(-1) = -36
\][/tex]
So, combining these results:
[tex]\[
-4i(5 - 9i) = -20i - 36
\][/tex]
### Combining both parts:
Now we subtract the second simplified expression from the first one:
[tex]\[
(8 - 20i) - (-20i - 36)
\][/tex]
Rewriting this, we get:
[tex]\[
8 - 20i + 20i + 36
\][/tex]
Simplifying further:
[tex]\[
8 + 36 = 44 \quad \text{(Real part)}
\][/tex]
[tex]\[
-20i + 20i = 0i \quad \text{(Imaginary part)}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{-28 + 0i}
\][/tex]
