Question

divide.
\\(\frac{5 - 4i}{-2 + i}\\)
write your answer as a complex number in

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $-2-i$ (the conjugate of $-2+i$):
$$\frac{5-4i}{-2+i} \times \frac{-2-i}{-2-i} = \frac{(5-4i)(-2-i)}{(-2+i)(-2-i)}$$

Step2: Expand numerator via FOIL

Calculate product of the numerator terms:
$$(5)(-2) + (5)(-i) + (-4i)(-2) + (-4i)(-i) = -10 -5i +8i +4i^2$$
Substitute $i^2=-1$:
$$-10 +3i +4(-1) = -10 +3i -4 = -14 +3i$$

Step3: Expand denominator (difference of squares)

Calculate product of the denominator terms:
$$(-2)^2 - (i)^2 = 4 - (-1) = 4 +1 =5$$

Step4: Simplify the fraction

Divide the expanded numerator by the denominator:
$$\frac{-14 +3i}{5} = -\frac{14}{5} + \frac{3}{5}i$$

Answer:

$-\frac{14}{5} + \frac{3}{5}i$