To solve this equation, we can let [tex]y = x^{2}[/tex].
Then, the equation becomes:

[tex]y^{2} - 3y - 4 = 0[/tex]

Factorizing this quadratic equation, we get:

[tex](y - 4)(y + 1) = 0[/tex]

Therefore, [tex]y = 4[/tex] or [tex]y = -1[/tex].

Now, substituting back [tex]y = x^{2}[/tex], we have:

[tex]x^{2} = 4[/tex] or [tex]x^{2} = -1[/tex]

So, [tex]x = \pm 2[/tex] or [tex]x = \pm i[/tex].

Therefore, the solutions to the equation [tex]x^{4} - 3x^{2} - 4 = 0[/tex] are [tex]x = \pm 2[/tex] or [tex]x = \pm i[/tex].