Question

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)

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:

\(-28 - 40i\)