Question

multiply. choose the correct answer from the choices below.
$(3 - 4i)(-2 + 5i)$
a $14 + 23i$
b $26 + 8i$
c $14 + 8i$
d $26 + 23i$

Explanation:

Step1: Use distributive property (FOIL)

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

Step2: Simplify each product

Calculate each product:
$$3\times(-2)= -6$$
$$3\times(5i)=15i$$
$$-4i\times(-2)=8i$$
$$-4i\times(5i)= -20i^{2}$$
So the expression becomes:
$$-6 + 15i + 8i - 20i^{2}$$

Step3: Recall that \(i^{2}=-1\)

Substitute \(i^{2}=-1\) into the expression:
$$-6 + 15i + 8i - 20\times(-1)$$
$$-6 + 15i + 8i + 20$$

Step4: Combine like terms

Combine the real parts and the imaginary parts:
Real parts: \(-6 + 20 = 14\)
Imaginary parts: \(15i + 8i = 23i\)
So the result is \(14 + 23i\)

Answer:

A. \(14 + 23i\)