Distribute to simplify the expression (3i + 4)(−2i − 5)
.(1 point)
To simplify the expression \((3i + 4)(-2i - 5)\), you need to distribute, or apply the distributive property, which states that for any three numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). Here we will use it to multiply two binomials.
Let's multiply each term from the first bracket with each term from the second bracket:
\[
(3i + 4)(-2i - 5) = 3i(-2i) + 3i(-5) + 4(-2i) + 4(-5)
\]
Now let's perform each multiplication:
\(3i(-2i) = -6i^2\)
Note that \(i^2 = -1\) (since \(i\) is the imaginary unit), so:
\(-6i^2 = -6(-1) = 6\)
Now we continue with the rest of the terms:
\(3i(-5) = -15i\)
\(4(-2i) = -8i\)
\(4(-5) = -20\)
Now combine all of these:
\[
(3i + 4)(-2i - 5) = 6 - 15i - 8i - 20
\]
Next, combine like terms (-15i and -8i):
\[
6 - 23i - 20
\]
Finally, combine the real numbers (6 and -20):
\[
6 - 20 = -14
\]
So the simplified form of the expression is:
\[
-14 - 23i
\]