Simplify the following quotient of complex numbers into the form

a + bi. -8-8i/1+2i

This can be accomplished by multiplying numerator and denominator by the complex conjugate of the denominator, 1-2i.

(-8-8i)*(1-2i) / ((1+2i)(1-2i))
8(-1-2 -i+2i) /(1-4i²)
8(i-3)/5

To simplify the quotient of complex numbers, we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

The conjugate of 1 + 2i is 1 - 2i.

So, multiplying the numerator and denominator by the conjugate, we have:

((-8 - 8i) * (1 - 2i))/((1 + 2i) * (1 - 2i))

Now, let's simplify the numerator:

((-8 - 8i) * (1 - 2i))
= (-8 * 1) + (-8i * 1) + (-8 * -2i) + (-8i * -2i)
= -8 - 8i + 16i + 16i^2

Since i^2 is equal to -1, we can substitute it in:

= -8 - 8i + 16i - 16
= -24 + 8i

Now, let's simplify the denominator:

(1 + 2i) * (1 - 2i)
= 1 - 2i + 2i - 4i^2
= 1 - 4i^2

Again, substituting i^2 with -1:

= 1 - 4(-1)
= 1 + 4
= 5

Now, we have:

(-24 + 8i)/5

So, the simplified form of the quotient is:

-24/5 + (8/5)i