all real and imaginary numbers and multiplicity for x^4+2x^2+1=0. Show work. Thanks!

Answer:x = i, mult 2; x = -i, mult 2Step-by-step explanation:First let's make a substitution to make this easier to factor.  Let u² = x⁴ and       u = x²Now we can rewrite the polynomial asu² +2u + 1 = 0This factors easily into(u + 1)(u + 1) = 0By the Zero Product Property, eitheru + 1 = 0 or u + 1 = 0Putting back the x²:x² + 1 = 0 or x² + 1 = 0For the first one, even though they are the same:x² = -1 sox = ±√-1 Since that is not "allowed", we make the replacement of -1 = i²:x = ±√i² sox = ±iFor the second one, we need not repeat the whole process, but we find 2 more identical roots:x = ±iThat means that the factors are(x + i)(x - i)(x + i)(x - i) = 0x = i, multiplicity 2 andx = -i, multiplicity 2