To multiply the complex numbers \( (5 - 10i)(-3 + 8i) \), we can use the distributive property (also known as the FOIL method for binomials):
\[ (5 - 10i)(-3 + 8i) = 5 \cdot (-3) + 5 \cdot (8i) - 10i \cdot (-3) - 10i \cdot (8i) \]
Calculating each term:
- \( 5 \cdot (-3) = -15 \)
- \( 5 \cdot (8i) = 40i \)
- \( -10i \cdot (-3) = 30i \)
- \( -10i \cdot (8i) = -80i^2 \)
Since \( i^2 = -1 \), we can simplify \( -80i^2 \):
\[ -80i^2 = -80(-1) = 80 \]
Now we combine all the terms:
\[ -15 + 40i + 30i + 80 = (-15 + 80) + (40i + 30i) = 65 + 70i \]
Thus, the result of multiplying \( (5 - 10i)(-3 + 8i) \) is:
\[ \boxed{65 + 70i} \]