This gives us roots of

The product of with each of these gives us:

The sum of these 4 is:

What we notice is that each of the roots has a negative. It thus makes sense that they will all cancel out. Rather than going through all the multiplication, we can instead look at the very beginning setup, which we can simplify using the distributive property:

Distribute (3 + 4i)(4 - 3i)

3(4) + 3(-3i) + 4i(4) + 4i(-3i)

12 - 9i + 16i -12i2

12 + 7i - 12(-1)

12 + 7i + 12

24 + 7i

Use FOIL:

Combine like terms:

But since , we know

Apply the Power of a Product Rule:

,

and

,

so, substituting and evaluating:

To raise any expression to the third power, use the pattern

Setting :

Taking advantage of the Power of a Product Rule:

Since ,

and

:

Collecting real and imaginary terms:

Use FOIL to multiply complex numbers as follows:

Since , it follows that , so then:

Combining like terms gives:

To raise any expression to the third power, use the pattern

Setting :

Taking advantage of the Power of a Product Rule:

Since ,

and

:

Collecting real and imaginary terms:

conforms to the perfect square trinomial pattern

.

The easiest way to solve this problem is to add and , then square the sum.

The complex conjugate of a complex number is .

,

so is the complex conjugate of this;

,

and

Substitute 8 for :

.

Apply the Power of a Product Rule:

Applying the Product of Powers Rule:

raised to any multiple of 4 is equal to 1, and , so, substituting and evaluating:

This is not among the given choices.

conforms to the perfect square trinomial pattern

.

The easiest way to solve this problem is to subtract and , then square the difference.

The complex conjugate of a complex number is .

,

so is the complex conjugate of this;

Taking advantage of the Power of a Product Rule and the fact that :