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: $$\boxed{{\displaystyle 65+70i}}$$