Question

simplify.
\\(\frac{-4}{-5 + 4i}\\)
write your answer in the form \\(a + bi\\). reduce all fractions.

Explanation:

Step1: Multiply by conjugate of denominator

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

Step2: Expand numerator and denominator

Calculate products and simplify $i^2=-1$:
$$\frac{20 + 16i}{25 - (-16)} = \frac{20 + 16i}{41}$$

Step3: Split into a + bi form

Separate real and imaginary parts:
$$\frac{20}{41} + \frac{16}{41}i$$

Answer:

$\frac{20}{41} + \frac{16}{41}i$