Question

Distribute to simplify the expression (−3i−5)[(i+2)+(2i−4)

__+__i

Answer 1

To simplify the expression \((-3i - 5)[(i + 2) + (2i - 4) + i]\), let's break it down step by step.

First, simplify the expression inside the brackets:
\[
(i + 2) + (2i - 4) + i
\]

Combine like terms:
1. Combine the \(i\) terms: \(i + 2i + i = 4i\)
2. Combine the constant terms: \(2 - 4 = -2\)

Thus, we have:
\[
(i + 2) + (2i - 4) + i = 4i - 2
\]

Next, substitute this back into the original expression:
\[
(-3i - 5)(4i - 2)
\]

Now, use the distributive property (also known as the FOIL method for binomials):
\[
(-3i)(4i) + (-3i)(-2) + (-5)(4i) + (-5)(-2)
\]

Calculating each term:
1. \(-3i \cdot 4i = -12i^2\)
2. \(-3i \cdot -2 = 6i\)
3. \(-5 \cdot 4i = -20i\)
4. \(-5 \cdot -2 = 10\)

Now we combine all the like terms:
\[
-12i^2 + 6i - 20i + 10
\]

Combine the \(i\) terms:
\[
-12i^2 - 14i + 10
\]

Since \(i^2 = -1\), we can simplify \(-12i^2\) to:
\[
-12(-1) = 12
\]

Thus, the expression becomes:
\[
12 - 14i + 10
\]

Finally, combine the constant terms:
\[
12 + 10 = 22
\]

So the final simplified expression is:
\[
22 - 14i
\]

In conclusion, the simplified expression is:
\[
\boxed{22 - 14i}
\]