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 \[\begin{align} & x^4+6x^3+17x^2-6x-18 \\ &= (x+a)(x-a)(x+b+ci)(x+b-ci), \end{align} \] 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 \( c > 0 \).

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