Question

1. (4 - 2i)(-3 + 5i)

Explanation:

Step1: Use distributive property (FOIL)

Multiply each term in the first complex number by each term in the second:
\( (4)(-3) + (4)(5i) + (-2i)(-3) + (-2i)(5i) \)

Step2: Simplify each product

Calculate each term:
\( 4 \times -3 = -12 \), \( 4 \times 5i = 20i \), \( -2i \times -3 = 6i \), \( -2i \times 5i = -10i^2 \)

Step3: Substitute \( i^2 = -1 \)

Replace \( i^2 \) with -1:
\( -10i^2 = -10(-1) = 10 \)

Step4: Combine like terms

Combine real parts and imaginary parts:
Real: \( -12 + 10 = -2 \)
Imaginary: \( 20i + 6i = 26i \)

Answer:

\( -2 + 26i \)