Question

1. \\(\frac{4 + i}{3 + 2i}\\)

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $3-2i$:
$$\frac{(4+i)(3-2i)}{(3+2i)(3-2i)}$$

Step2: Expand numerator using FOIL

Calculate product of binomials:
$$(4)(3) + (4)(-2i) + (i)(3) + (i)(-2i) = 12 - 8i + 3i - 2i^2$$
Simplify using $i^2=-1$:
$$12 - 5i - 2(-1) = 12 - 5i + 2 = 14 - 5i$$

Step3: Expand denominator (difference of squares)

Use $(a+b)(a-b)=a^2-b^2$:
$$3^2 - (2i)^2 = 9 - 4i^2 = 9 - 4(-1) = 9 + 4 = 13$$

Step4: Write simplified fraction

Divide simplified numerator by denominator:
$$\frac{14 - 5i}{13} = \frac{14}{13} - \frac{5}{13}i$$

Answer:

$\frac{14}{13} - \frac{5}{13}i$