solve the following equation in the complex number system Please show all your work.

Question

5x^5 + 32x^4 - 5x - 32 = 0

Answer 1

A little synthetic division shows

5x^5 + 32x^4 - 5x - 32
= (x-1)(x+1)(5x^3 + 32x^2 + 5x + 32)
Notice the 5,32 5,32 coefficients.
a little regrouping gives

(x-1)(x-1)(x^2(5x+32) + (5x+32))
(x-1)(x+1)(x^2 + 1)(5x+32)

Actually, we could have saved ourselves the synthetic division by noticing the 5,32,5,32 grouping at first:

(x^4 - 1)(5x+32)

You can probably take it from here . . .