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.