Mutiply (5-10i)(-3+8i) provide answer simplified to the finest and for a+bi

Answer 1

To multiply the complex numbers \( (5 - 10i)(-3 + 8i) \), we will use the distributive property (also known as the FOIL method for binomials).

1. First: Multiply the real parts: \[ 5 \cdot -3 = -15 \]

2. Outer: Multiply the outer parts: \[ 5 \cdot 8i = 40i \]

3. Inner: Multiply the inner parts: \[ -10i \cdot -3 = 30i \]

4. Last: Multiply the imaginary parts: \[ -10i \cdot 8i = -80i^2 \] Since \( i^2 = -1 \), this becomes: \[ -80(-1) = 80 \]

Now combine all these results: \[ -15 + 40i + 30i + 80 \]

Combine like terms: \[ (-15 + 80) + (40i + 30i) = 65 + 70i \]

So, the product \( (5 - 10i)(-3 + 8i) \) simplifies to: \[ \boxed{65 + 70i} \]