To multiply the complex numbers \( (5 - 10i)(-3 + 8i) \), we can use the distributive property (also known as the FOIL method for binomials):
\[ (5 - 10i)(-3 + 8i) = 5 \cdot -3 + 5 \cdot 8i - 10i \cdot -3 - 10i \cdot 8i \]
Now, we calculate each term:
- \( 5 \cdot -3 = -15 \)
- \( 5 \cdot 8i = 40i \)
- \( -10i \cdot -3 = 30i \)
- \( -10i \cdot 8i = -80i^2 \)
Since \( i^2 = -1 \), we replace \( -80i^2 \) with \( 80 \):
Combining all the terms together:
\[ -15 + 40i + 30i + 80 \]
Combine the real parts and the imaginary parts:
Real part: \( -15 + 80 = 65 \)
Imaginary part: \( 40i + 30i = 70i \)
Therefore, the result of the multiplication is:
\[ 65 + 70i \]