In a monic quadratic with roots r and s we have
f(x) = (x - r)(x - s) = x² - (r+s) x + rs
Here we have r = 5 - 3i and s must be the conjugate to get real coefficients,
s = 5 + 3i
r + s = 5 - 3i + 5 + 3i = 10
rs = rr* = |r|² = 5²+3² = 34
Answer: x² - 10x + 34 = 0
