type the correct answer in the box. simplify the following expression into the form a + bi, where a and b are rational numbers. (3 + 7i)(-2 - 2i) - 4i(5 - 9i)

Question

Accepted Answer

# Explanation:
## Step1: Expand the first product $(3 + 7i)(-2 - 2i)$
Using the distributive property (FOIL method):
$$
\begin{align*}
(3 + 7i)(-2 - 2i)&=3\times(-2)+3\times(-2i)+7i\times(-2)+7i\times(-2i)\
&=-6 - 6i - 14i - 14i^{2}
\end{align*}
$$
Since $i^{2}=-1$, we have:
$$
\begin{align*}
-6 - 6i - 14i - 14i^{2}&=-6 - 20i - 14\times(-1)\
&=-6 - 20i + 14\
&=8 - 20i
\end{align*}
$$

## Step2: Expand the second product $4i(5 - 9i)$
Using the distributive property:
$$
\begin{align*}
4i(5 - 9i)&=4i\times5 - 4i\times9i\
&=20i - 36i^{2}
\end{align*}
$$
Since $i^{2}=-1$, we have:
$$
\begin{align*}
20i - 36i^{2}&=20i - 36\times(-1)\
&=20i + 36\
&=36 + 20i
\end{align*}
$$

## Step3: Subtract the two expanded expressions
We have $(3 + 7i)(-2 - 2i)-4i(5 - 9i)=(8 - 20i)-(36 + 20i)$
$$
\begin{align*}
(8 - 20i)-(36 + 20i)&=8 - 20i - 36 - 20i\
&=(8 - 36)+(-20i - 20i)\
&=-28 - 40i
\end{align*}
$$

# Answer:
$-28 - 40i$

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QUESTION IMAGE

Question

Explanation:

Step1: Expand the first product \((3 + 7i)(-2 - 2i)\)

Step2: Expand the second product \(4i(5 - 9i)\)

Step3: Subtract the two expanded expressions

Answer: