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Explanation:
The given zeros are: 4, -8, 2+3i
All coefficients are real numbers, so this means that the root 2+3i must be paired with 2-3i which is its conjugate pair.
While we are given 3 roots, there are actually four roots and they are: 4, -8, 2+3i, 2-3i
This means,
x = 4 or x = -8 or x = 2+3i or x=2-3i
x-4=0 or x+8=0 or x-2=3i or x-2=-3i
x-4=0 or x+8=0 or (x-2)^2=(3i)^2 or (x-2)^2=(-3i)^2
x-4=0 or x+8=0 or (x-2)^2=-9 or (x-2)^2=-9
x-4=0 or x+8=0 or (x-2)^2=-9
x-4=0 or x+8=0 or (x-2)^2+9=0
(x-4)(x+8)((x-2)^2+9)=0
(x-4)(x+8)(x^2-4x+4+9)=0
(x-4)(x+8)(x^2-4x+13)=0
(x^2+4x-32)(x^2-4x+13)=0
x^4-35x^2+180x-416 = 0 .... see attached image below
If you work your way backwards up the steps listed here, then you'll go from x^4-35x^2+180x-416 = 0 to x = 4 or x = -8 or x = 2+3i or x=2-3i
This shows f(x) = x^4-35x^2+180x-416 has the four roots x = 4, x = -8, x = 2+3i, x = 2-3i
