ANSWER
[tex]x = \: x = \pm \: \sqrt{2} i \: or \: x = \pm \: i[/tex]
EXPLANATION
[tex] {x}^{4} + 3 {x}^{2} + 2 = 0[/tex]
[tex]{ ({x}^{2}) }^{2} + 3( {x}^{2}) + 2 = 0[/tex]
Let
[tex]u = {x}^{2} [/tex]
Then the equation becomes:
[tex] {u}^{2} + 3u + 2 = 0[/tex]
[tex] {u}^{2} + 3u + 2 = 0[/tex]
[tex] {u}^{2} + 2u +u + 2 = 0[/tex]
Factor:
[tex]{u}(u + 2)+ 1(u + 2) = 0[/tex]
[tex](u + 1)(u + 2) = 0[/tex]
[tex]u = - 1[/tex]
or
[tex]u = - 2[/tex]
This implies that
[tex] {x}^{2} = - 1 \implies \: x = \pm \: i[/tex]
or
[tex] {x}^{2} = - 2 \implies \: x = \pm \: \sqrt{2} i[/tex]
