Question

express as a complex number in simplest a+bi form:
\frac{-19 - 22i}{1 + 8i}

Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by $1-8i$ (the conjugate of $1+8i$):
$$\frac{-19-22i}{1+8i} \times \frac{1-8i}{1-8i} = \frac{(-19-22i)(1-8i)}{(1+8i)(1-8i)}$$

Step2: Expand numerator

Use FOIL method for numerator:
$$(-19)(1) + (-19)(-8i) + (-22i)(1) + (-22i)(-8i) = -19 + 152i -22i + 176i^2$$
Substitute $i^2=-1$:
$$-19 + 130i + 176(-1) = -19 -176 +130i = -195 +130i$$

Step3: Expand denominator

Use difference of squares:
$$1^2 - (8i)^2 = 1 - 64i^2$$
Substitute $i^2=-1$:
$$1 - 64(-1) = 1 +64 = 65$$

Step4: Simplify fraction

Divide each term by 65:
$$\frac{-195}{65} + \frac{130}{65}i = -3 + 2i$$

Answer:

$-3 + 2i$