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

Answer 1

To simplify the expression \((3i + 4)(-2i - 5)\), we will use the distributive property (also known as the FOIL method for binomials).

1. First: Multiply the first terms: \[ 3i \cdot (-2i) = -6i^2 \]

2. Outer: Multiply the outer terms: \[ 3i \cdot (-5) = -15i \]

3. Inner: Multiply the inner terms: \[ 4 \cdot (-2i) = -8i \]

4. Last: Multiply the last terms: \[ 4 \cdot (-5) = -20 \]

Now, combine all these results: \[ -6i^2 - 15i - 8i - 20 \]

Combine like terms (\(-15i\) and \(-8i\)): \[ -6i^2 - 23i - 20 \]

Since \(i^2 = -1\), we can substitute that in: \[ -6(-1) - 23i - 20 = 6 - 23i - 20 \]

Combine the real parts: \[ (6 - 20) - 23i = -14 - 23i \]

So, the simplified expression is: \[ \boxed{-14 - 23i} \]