How am I supposed to solve this equation?

Question

Find an nth-degree polynomial function with real coefficents satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; 2 and 4i are zeros; f(1)=-34

Answer 1

zeros 2 and 4i, also -4i since imaginary zeros come in conjugate pairs

x=2

x-2=0

x-2 is a factor

x=4i

x^2 =-16

x^2+16 is another factor

multiply the 2 factors together

a(x-2)(x^2+16)

= a(x^3 -2x^2 +16x -32 ) = f(1)=-34

=a(1-2+16-32) =-34

-17a = -34

a =-34/-17 = 2

f(x) = 2(x^3-2x^2 +16x-32)

f(x) = 2x^3 -4x^2 +32x -64