Question

The roots of x2 − () + 34 are 5 ± 3i.

Answer 1

What do you mean by " () "   ?

Let's check the roots and see what middle term results.  We must multiply here:

If 5+3i is a root, then x-5-3i is a factor.  Also, if 5-3i is a root, then x-5+3i is a factor.

We must multiply together (x-5-3i)(x-5+3i).

One way of doing this is as follows:  [(x-5) + 3i][(x-5) - 3i]

Recalling that (a+b)(a-b) = a^2 - b^2,

                 [(x-5) + 3i][(x-5) - 3i] becomes (x+5)^2 + 9, which
can be expanded and simplified as follows:   

                        x^2 + 10x + 25 + 34

So, your   " () " in your  "x^2 − () + 34" should be "10x + 25."

In summary, the roots of x^2 + 10x + 25 + 34 are 5+3i and 5-3i.