Algebra
# Complex Numbers

If $x (x-4-9i)(x-a+bi) = x^3 -8x^2+97x,$ what is $a + b$ ?

**Details and assumptions**

$i$ is the the imaginary unit, defined by $i^2 = -1$.

Suppose that $\begin{aligned} & x^4+6x^3+17x^2-6x-18 \\ &= (x+a)(x-a)(x+b+ci)(x+b-ci), \end{aligned}$ where $i$ is the imaginary number that satisfies $i^2=-1$. What is the value of $a+b+c$?

**Details and assumptions**

Assume $a > 0$ and $c > 0$.