Complex roots of polynomial equations with real coefficients Consider the polynomial $x^5 +ax^4 +bx^3 +cx^2 +dx+4$ where $a, b, c, d$ are real
numbers. If $(1 + 2i)$ and $(3 - 2i)$ are two roots of this polynomial then what is the value of $a$ ?

As the coefficients of the different powers of $x$ are real, all the complex roots must occur with conjugate pair.
So, the other two roots are $1-2i,3+2i$ 
So if $p$ is the fifth root,
using Vieta's Formulas $(1-2i)(3+2i)(1+2i)(3-2i)p=(-1)^5\frac41$
and $(1-2i)+(3+2i)+(1+2i)+(3-2i)+p=-\frac a1$
