Question

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

Answer 1

To simplify the expression \((−3i−5)[(i+2)+(2i−4)]\), we first simplify the expression inside the brackets.

1. Combine the terms inside the brackets:
\[
(i + 2) + (2i - 4) = i + 2 + 2i - 4 = (i + 2i) + (2 - 4) = 3i - 2.
\]

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

2. Next, we distribute \((-3i - 5)\) with \((3i - 2)\):
\[
(-3i)(3i) + (-3i)(-2) + (-5)(3i) + (-5)(-2).
\]

Calculating each term:
- \((-3i)(3i) = -9i^2\)
- \((-3i)(-2) = 6i\)
- \((-5)(3i) = -15i\)
- \((-5)(-2) = 10\)

Now combine these results:
\[
-9i^2 + 6i - 15i + 10.
\]

This simplifies to:
\[
-9i^2 - 9i + 10.
\]

3. Since \(i^2 = -1\), replace \(-9i^2\) with \(9\):
\[
9 - 9i + 10.
\]

4. Finally, combine like terms:
\[
9 + 10 - 9i = 19 - 9i.
\]

Thus, the simplified expression is:
\[
\boxed{19 - 9i}.
\]