Answer:
x = i, mult 2; x = -i, mult 2
Step-by-step explanation:
First let's make a substitution to make this easier to factor.
Let u² = x⁴ and
u = x²
Now we can rewrite the polynomial as
u² +2u + 1 = 0
This factors easily into
(u + 1)(u + 1) = 0
By the Zero Product Property, either
u + 1 = 0 or u + 1 = 0
Putting back the x²:
x² + 1 = 0 or x² + 1 = 0
For the first one, even though they are the same:
x² = -1 so
x = ±√-1
Since that is not "allowed", we make the replacement of -1 = i²:
x = ±√i² so
x = ±i
For the second one, we need not repeat the whole process, but we find 2 more identical roots:
x = ±i
That means that the factors are
(x + i)(x - i)(x + i)(x - i) = 0
x = i, multiplicity 2 and
x = -i, multiplicity 2
