Question

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

Explanation:

Step1: Apply the distributive property (FOIL method)

We multiply each term in the first complex number by each term in the second complex number:
$$(1 - 2i)(3 + 4i)=1\times3+1\times4i-2i\times3-2i\times4i$$

Step2: Simplify each term

Calculate the products:
$$1\times3 = 3$$, $$1\times4i=4i$$, $$-2i\times3=-6i$$, $$-2i\times4i=-8i^{2}$$
So the expression becomes:
$$3 + 4i-6i-8i^{2}$$

Step3: Combine like terms and use \(i^{2}=- 1\)

Combine the imaginary terms \(4i-6i=-2i\), and substitute \(i^{2}=-1\) into \(-8i^{2}\):
$$3-2i-8\times(-1)=3 - 2i + 8$$

Step4: Combine the real terms

Combine the real numbers \(3 + 8 = 11\):
$$11-2i$$

Answer:

\(11 - 2i\)